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Full text of "Empirical investigation into the behavior of options around merger and acquisition announcements"

AN EMPIRICAL INVESTIGATION INTO THE BEHAVIOR OF OPTIONS 
AROUND MERGER AND ACQUISITION ANNOUNCEMENTS 



By 
JAMES A. YODER 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 

OF THE UNIVERSITY OF FLORIDA IN 

PARTIAL FULFILLMENT OF THE REQUIREMENTS 

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 

UNIVERSITY OF FLORIDA 

1988 



OF F LIB 



ACKNOWLEDGEMENTS 

I would like to express my special thanks to the chairman of my 
committee, Haim Levy, and to Drs. Roger Huang, Roy Crum and Sandy Berg. 
T would also like to express my appreciation to Drs. Andy McCollough, 
Craig Tapely, Robert Radcliffe, Dave Brown and Joel Houston for their 
encouragement and support. 

I would also like to express my gratitude to my fellow students 
Young Hoon Byun, Lesa Nix, Bruce Kuhlman and Neil Sicherman for their 
suggestions and technical assistance. I would also like to acknowledge 
the computer programming assistance of Eric Olson. 



11 



TABLE OF CONTENTS 

Page 

ACKNOWLEDGMENTS . . a 

ABSTRACT v 

CHAPTERS 

1 INTRODUCTION 1 

The Behavior of Option Prices Around Merger Announcements 2 
The Behavior of Implied Standara Deviations Around Merger 

Announcements 4 

Does the Option Market React to Merger/Acquisition 

Activity Differently than the Equity Market? 5 

How Does an Event Study in the Option Market Differ From 

One in the Equity Marker? ......... 8 

2 REVIEW OF THE LITERATURE 12 

Mergers 12 

Options 16 

Pricing 16 

Option Market Efficiency 18 

Variance Bias in the Black-Scholes Model 19 

3 THE BEHAVIOR OF OPTIONS AND OPTION MARKETS AROL^ND MERGER 

AND ACQUISITION ANNOUNCEMENTS ... 20 

Data 20 

The Behavior of Options Around Merger and Acquisition 

Announcements 22 

The Behavior of ISDs Around Merger and Acquisition 

Announcments 24 

Methodology ..... 26 

Interpretation Results 29 

The Behavior of Call Option Prices Around Merger and 

and Acquisition Announcements 34 

Methodology and Results 36 

Interpretation Results 40 

The Behavior of Options Around Merger and Acquisition 

Announcements 43 

Methods and Results 47 

Methods and Results 50 



1X1 



4 VAEIANCE BIAS AND NON- SYNCHRONOUS PRICES IN THE BLACK- 

SCHOLES MODEL 60 

Variance Bias in the Black-Scholes Model 61 

Methodology and Results 64 

Non- synchronous Prices and the Black-Scholes Model .... 71 

Methodology and Results 72 

Conclusion 77 

5 SUMMARY AND CONCLUSIONS 79 

REFERENCES 83 

BIOGRAPHICAL SKETCH 87 



IV 



Abstract of Dissertation Presented to the Graduate School 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 



AN EMPIRICAL INVESTIGATION INTO THE BEHAVIOR OF OPTIONS 
AROUND MERGER AND ACQUISITION ANNOUNCEMENTS 

By 

James A. Yoder 
December 1988 

Chairman: Haim Levy 

Major Department: Finance, Insurance, and Real Estate 

This dissertation exam.ines the behavior of options around merger 

and acquisition announcements. A variation of the traditional event 

study methodology was applied to the option market in order to measure 

the abnormal returns accruing to the bidding firm and target firm 

optionholders. The event study was extended to the equity market for 

comparison purposes. The behavior of ISDs was also examined in order 

to determine whether the option or equity market first reacted to the 

merger/acquisition announcement and to decompose the abnormal returns 

in the option market into a component due to changing stock prices and 

a component due to changing stock volatilities. Some methodological 

issues involving event studies were also examined. 



CHAPTER 1 
INTRODUCTION 

Two of the most important developments in finance in recent years 
have been the growth of option markets and the high level of merger and 
acquisition activity. Not surprisingly, both of these areas have been 
subject to intense academic scrutiny. Literally hundreds of articles 
have been published on the theory and applications of options. There 
are also numerous papers concerned with the rationale for mergers and 
their impact on stockholders' wealth. This dissertation attempts to 
relate these two subjects through an examination of oprion and option 
market behavior around merger and acquisition announcements. 

In order to accomplish this, four major issues will be addressed: 

1. How do option prices react around merger/acquisition 
announcements ? 

2. How do the Implied Standard Deviations (ISDs) of options 
react around merger and acquisition announcements? 

3. Does the option market react to merger and acquisition 
activity differently than the equity market? 

A. How does an event study in the option market differ from one 
in the equity market? 
Each of these issues will be discussed in turn. 



The Behavior of Option Prices 
Around Merger Announcements 

Researchers in the equity market have sought to determine whether 
mergers and acquisitions produce economic gains and, if so, who reaps 
the benefits. Their findings have been relatively consistent. Dodd 
(1980), Asquith (1983) and Eckbo (1983), for example, have all 
presented evidence on the effects of mergers on shareholders' wealth. 
They conclude that most of the gains are captured by the stockholders 
of the target firm. Gains to the bidding firm shareholders are small 
and possibly non-existent. Their estimates of the abnormal returns 
accruing to the bidding firm shareholders for the two days prior to the 
announcement range from a -1.09 percent loss to a paltry 0.20 percent 
gain. For the target firm shareholders, however, statistically signif- 
icant gains ranging from 6.20 percent to 13.41 percent were obtained. 
The merger literature is discussed more thoroughly in Chapter 2. 

These results in the equity market lead to empirically testable 
hypotheses for the expected behavior of options around merger and 
acquisition announcem.ents . Under the assumption that the option market 
!s efficient, option prices (and ISDs) can be expected to react prior 
to the formal merger announcement and stabilize immediately afterward. 
Merger negotiations involve many people such as investment bankers, 
lawyers, administrative personnel, etc. Word of impending mergers 
:!eaking to the financial market place has been amply demonstrated in 
the equity market. There is no reason why the same phenomenon should 
not occur in the option market. 

A second hypothesis is that abnormal returns to the target firm 
optionholders should exceed those of the bidding firm optionholders. 
Theoretically, a call option can be duplicated by an appropriately 



selected stock-bond portfolio. Because of this, the wealth effects of 
merger and acquisition announcements on optionholders can be expected 
to mirror that of the equityholders. 

The wealth effects of merger and acquisition activity on option- 
holders is of interest for a number of reasons. In a recent survey 
article of the market for corporate control literature, Jensen and 
Ruback (1983) identified six key questions that have been addressed. 
At the top of the list is the following = "How large are the gains to 
shareholders of bidding and target firms?" Options by their very 
nature afford superior leverage to the underlying equity. Consequently, 
optionholders, per dollar invested, have more reason to be concerned 
with potential merger and acquisition activity than the equityholders. 
An analysis of option prices around merger and acquisition announce- 
ments may also shed light on a puzzling question. 

Merger activity is widespread but the rationale for it is not 
clear. As noted earlier, gains to the bidding firm shareholders are 
small and possibly negative. Why then do managers undertake merger and 
acquisition programs if they do not benefit the shareholders? An 
examination of bidding firm option prices may help to resolve this 



issue, 



An option can be interpreted as a leveraged position in the 
equity. This leverage aspect of options may make them more sensitive 
to events than the underlying equity. Small abnormal returns in the 
equity market might result in much larger abnormal returns in the 
option market. Thus, it may be easier from a statistical standpoint to 
determine if bidding firm stockholders benefit from merger/acquisition 
activity by looking at the behavior of associated option prices. 



The Behavior of Implied Standard 
Deviations Around Merger Announcements 

The behavior of ISDs around merger announcements is important for 
two reasons. First, it is inseparable from the price behavior of 
options. Stock volatility is one of the input variables for the Black- 
Scholes model. By examining the ISD, it is possible to decompose 
changes in option prices into a component due to price changes in the 
underlying stock and a component due to changes in the underlying 
volatility. Second, it provides an alternative measure of the informa- 
tion content associated with merger and acquisition announcements. 
Numerous studies have attempted to measure the information content of 
accounting announcements by showing that the expected return of the 
stock is affected. Patell and Wolfson (1979, 19Sl) have pointed out 
that other moments of stock price distribution may also be affected by 
the announcement and thus serve as a measure of its significance. They 
proceeded to use ISDs as an ex-ante measure of the information content 
associated with earnings announcements whose disclosure date is known. 
This dissertation attempts to use ISDs to measure the expected impact 
of merger and acquisition announcements which are totally or at best 
partially anticipated events. 

The hypotheses concerning the behavior of ISDs around merger and 
acquisition announcements parallels that of option prices. The first 
hypothesis is that ISDs should react prior to the formal announcement 
and stabilize immediately afterward. The second hypothesis is that the 
change in ISDs for the target firm should exceed that of the bidding 
''irm options. Merger and acquisition activity has very little impact 
on the bidding firm shareholders. Thus, the distribution of stock 



returns should not significantly change as a result of the announce- 
ment. The target firm shareholders, however, are greatly affected by 
the merger and acquisition activity. Increased volatility of the 
underlying stock returns can be generated by a multitude of factors. 
Uncertainty, for example, can arise over the anticipated terms of the 
agreement, whether a competing offer will be made or even whether the 
deal will be consummated. 

Does the Option Market React to Merger and Acquisition 
Activity Differently than the Equity Market? 

The third major area of inquiry in this dissertation is the 

relationship between the option and equity markets around merger 

announcements. There are two independent arguments for hypothesizing 

that the merger activity will be more strongly manifested in the option 

rather than the equity market prior to the announcement. 

Options represent leveraged positions in the underlying equity. The 

beta of an option in the Black-Scholes framework is always greater than 

that of the stock. Because of this, the option market may be more 

sensitive to events than the equity market. Even though the two 

markets may be reacting to the same information, the signal may first 

be more apparent and stronger in the option market. 

It is also possible that the option market contains information 

that is not incorporated in the equity market prior to mergers. As 

mentioned previously, a call or put option can be duplicated by an 

appropriate stock-bond portfolio. Because of this, options have been 

viewed as "derivative" assets whose prices are completely determined by 

the underlying equity. The possibility that the option market may 

influence the equity market has received little attention. It is 



conceivable that information is first processed in the option market 
and then filters to the equity market. A similar issue has been 
studied by Manaster and Rendleman (1982). They have advanced the 
intriguing hypothesis that the option market may play a key role in 
determining equilibrium stock prices. They argue that some investors 
may prefer to invest in the option rather than the equity market 
because of reduced transaction costs, fewer short selling restrictions, 
and most importantly, superior leverage. These traders could push 
option prices out of equilibrium relative to the underlying stocks. 
Arbitrageurs would then intervene to restore equilibrium between the 
two markets. 

Manaster and Rendleman attempted to test their theory. They 
"inverted" the Black-Scholes model to solve for the implied stock 
price. The implied stock price was then used to predict future stock 
prices. They found some evidence that the option market contains 
information that is not incorporated in the equity market. Unfor- 
tunately, their results are very weak and fatally flawed by their 
reliance on non- synchronous data. The data used in this dissertation 
will avoid this problem. 

In retrospect, Manaster and Rendlemans ' lack of results is not 
surprising. Both the option and equity markets react to public 
information. Generally, one would expect both markets to adjust 
simultaneously to new public information. On any given day for any 
particular corporation there may not be and probably is not information 
that is not fully reflected in both markets. 

However, this may not be true prior to major announcements by 
corporations such as mergers. In the case of mergers, the option 



market could be expected to be particularly influential in determining 
stock prices around merger announcements. Keown and Pinkerton (1981) 
have argued that information concerning impending mergers is suscep- 
tible to insider exploitation due to the large number of people 
typically involved in the negotiating process. They have attributed 
increased trading volume before the merger announcement to insider 
activity. An insider attempting to profit from knowledge of an impend- 
ing merger would have an incentive to use options because of their 
leverage aspects. To quote Fischer Black (1975, p. 61), "Since an 
investor can usually get more action from a given investment in options 
than he can by investing in the common stock, he may choose to deal 
with options when he feels he has an especially important piece of 
information." Option prices can be expected to contain more informa- 
tion than the equity market if nonpublic information is being 
exploited. If information regarding future mergers first reaches the 
financial markets through insider trading in options, merger activity 
may very well be reflected in the option market before the stock 
market. 

A separate issue raised by the above argument is that the option 
market m.ay make the equity market more efficient. If the option market 
serves to bring nonpublic information into the financial markets and 
options influence the prices of the underlying stock, then stocks with 
listed options should respond sooner to impending m.ergers than similar 
stocks without options. 



How Does an Event Study in the Option Market 
Differ From One in the Equitv Market? 



An event study attempts to measure the impact of some event on 
securityholders by comparing actual returns around the announcement to 
those predicted by some model. These predicted returns should be the 
returns that would have occurred if the event (merger and acquisitions 
in this case) had not taken place. The difference between the actual 
and predicted returns is the basic measure of the impact of the event. 
These residuals are then aggregated to measure the total impact of the 
event and provide statistics for tests of significance. 

In the equity market, predicted returns are usually generated by 
one of two models: 

a. Market model 

b. Mean returns 

The market model assumes there is a linear relationship between 
individual security returns and market returns: 

Kit = a^ + 3iRmt 

Where R^^ = return on company i for day t 
-R-nit = return on the market for day t 

The coefficients a and are obtained by regressing the company 
returns against the market returns over some base period prior to the 
merger activity. The base period should be selected so that the 
company returns are not affected by the event activity. 

.'\nother approach to generating predicted returns is to simply use 
the mean returns on the individual security computed over some base 
period 



_ n 

R^ = 1/N 2 R,;t 



where R^ = mean return on company i 

Rj_f = return of company i for day t 

N = number of observations in the base period 

Again the base period should be selected so that the event activity has 
no effect. 

Two implicit assumptions underlie the traditional event study 
methodology in the equity market. The first is that the return 
generating process is linear. As long as predicted returns equal actual 
returns on average, the residuals should average out to zero over a 
large enough cross-sectional sample in the absence of some conimon 
disturbing event. The same reasoning justifies parameter estimation 
for the two models. The true beta is unknown and must be estimated. 
The estimated beta may lie above or below the true value. As long as 
an unbiased of beta is used, however, deviations from the true beta 
return will average out to zero. Since these models are linear, 
deviations from the true expected return will also offset and residuals 
should average out to zero in the absence of a common disturbance. The 
second assumption is that the return generating process is stationary. 
Specifically, beta is assumed to remain constant over time. 

Call prices in a Black-Scholes framework are a function of five 
input variables. Two of these, the stock price and its volatility, are 
company specific and would be affected by an event such as a merger 
acquisition announcement. One implication of this is that there may be 



10 



a subtle but important difference between the interpretation of the 
results of an event study in the option and equity markets. 

i-!e.tger activity may not benefit a stockholder even if abnormal 
returns are observed. These abnormal returns may be accompanied by 
increased risk. This increased risk may not be desired by an investor 
with a small portfolio even if it is compensated for by larger expected 
(not realized) returns. 

If an investor holds a call option, the situation may be entirely 
different. An increase in the volatility of the underlying srock would 
definitely be preferred by all investors. Increased volatility would 
result in an actual (not expected) increase in the call price. The 
reason for this is that the return generating process underlying call 
prices is based on the formation of risk-less hedged portfolios. 

The Black-Scholes formula is by far the most widely used option 
pricing model. Using it to generate predicted returns for an event 
study, however, presents some technical problems. The Black-Scholes 
model is highly non-linear. Consequently, using sample estimates for 
the input variables may result in a systematic bias. Errors in estimat- 
ing the variables may offset in a large sample. Equal deviations from 
the true parameter estimate, however, will not result in equal devia- 
tions from the true call price. The most crucial variable is the stock 
volatility since the Black-Scholes model is most sensitive to it. 

Because of this problem, the results of an event study utilizing 
the Black-Scholes model must be interpreted with care. A simulation 
analysis, however, provides some m.easure of the magnitude of this 
effect. The Black-Scholes formula was used to generate a theoretical 
option price assuming true values for the input parameters. Sample 



11 



estimates of the volatility were generated for input into the Black- 
Scholes model. These sample call prices based on sample estimates for 
the volatility were compared to the theoretical call value. In 
general, the difference was small (see chapter A). 



CHAPTER 2 
REVIEW OF THE LITERATURE 



The literature on option theory and mergers, as mentioned previ- 
ously, is immense. It is impossible to discuss in detail all the 
relevant studies in either of these fields. At best, the most impor- 
tant results can only be highlighted. This section will give a brief 
review of the work that directly affects this dissertation. The 
literature dealing with the impact of mergers on shareholders wealth, 
option pricing, option market efficiency, and variance bias in the 
Black-Scholes model will be addressed in turn. 



Mergers 



Two fundamental questions have been raised regarding merger 
activity. The first is why do mergers occur? In 1985 alone, merger 
activity involved over $120 billion in assets. Yet the economic 
justification for all this activity is not obvious. Levy and Sarnat 
(1970) have shown that given perfect capital markets, pure conglomerate 
mergers should not create value. 

Agency theory provides one rationale for the continuous merger 
activity that has been observed over the past few decades. Levy and 
Sarnat (1970), Lewellen (1971) and Gali and Masulis (1976) have argued 
that combining firms with less than perfectly correlated cash flows 
lowers the chances for bankruptcy. Thus, managers have an incentive to 
engage in merger activity so as to reduce their employment risk. Reid 
(1968) has argued that managers strive to maxim.ize the size of the firm 

12 



13 

rather than shareholder wealth. Jensen and Heckling (1976) have pointed 
that since managers are agents for the stockholders' their interests 
are not necessarily the same. 

Others have sought to justify merger activity on the grounds that 
it produces real economic gains. Mergers may result in more efficient 
economic units. Weston and Chung (1983) have summarized possible 
sources of these efficiencies. 

1. Differential Efficiency 

2. Inefficient Managem.ent (target firm) 

3. Operating Synergy 
A. Financial Synergy 

5. Strategic Realignment 

6. Undervaluation (target firm) 

-..- of now, however, the exact rationale for mergers is still an 
unresolved issue. 

The second major issue that the merger literature has addressed is 
what are the effects of mergers on shareholders' wealth? Numerous 
studies concerned with this issue have appeared since Mandleker's 
(1974) seminal paper. Most of these have used the well known event 
study methodology. 

Event studies in the equity market involving mergers have become 
relatively standardized. A base period prior to the event is selected, 
and data from this period are used to estimate predicted returns. The 
impact of the event is measured by calculating the difference between 
the actual and predicted returns during some period surrounding the 
event. The residuals are then aggregated and statistically analyzed, 



14 



usually using some form of a t-test, to determine if the excess returns 
are significantly different from zero. 

Predicted returns in the equity market have usually been generated 
by one of two models. The first method is to use the market model. 

Rjt = aj + Bj"R^t 

The estimates of aj and Bj are obtained by regressing the company 
returns against the market returns during some base period prior to and 
presumably untainted by the merger activity. The second method is to 
simply use the mean return computed over some base period. 

Brown and Warner (1985) have shown that the event study metho- 
dology is very robust to the method used to calculate excess returns. 
Using simulation analysis, they showed that there is very little 
difference in the returns (or residuals) generated by the two methods. 
Because of this, mean adjusted returns will be used in this study. 

Jensen and Ruback (1983) have summarized the results of the more 
important merger studies concentrating on announcement effects. These 
results are shown in the table below. The top panel shows the results 
for the two days prior to the announcement. The bottom panel shows the 
results for the one month prior to the announcement. In each case, the 
total return during the event period, the number of observations and 
the t-statistic is given for both the bidding and target firms. 

The results are very consistent. The gains to the acquiring 
firms are positive but small. The target firm stockholders reap much 
larger returns. This is true for both the short-term (2 day) and long- 
term (one month) event period. In addition, these results hold for 
both successful (consummated) and ultimately unsuccessful mergers. 



15 



Table 2.1 



A'Dnormal Returns Associated with Mergers; 
Sample Size and t-statistic 



Study Sample Bidding firm Target firm 

period 



A. Two-day announcement effects 



T)odd /0-77 -1.09" +13.41 

(1980) (60"", -2. 98""'") (71^23.80) 

Asquith 62-76 +0.20 +6.20 

(1983) (196,0.78) (211,23.07) 

Eckbo 63-78 +0.07 +6.24 

(1983) (102,-0.12) (57,9.97) 

Weighted excess return -0.05 +7.72 



B. One-month announcement effects 

Eodd 70-77 +0.80 +21.78 

(60,0.67) (71,11.93) 

Asquith 62-76 +0.20 +13.30 

(1983) (196,0.25) (211,15.65) 

Eckbo 63-78 +1.58 +14.08 

(1983) (102,1.48) (57,6.97) 

Asquith et. 63-79 +3.48 +20.5 

al. (1983) (170,5.30) (35,9.54) 

Malatesta 69-74 +0.90 +16.8 

(1983) (256,1.53) (83,17.57) 

weighted excess return +1.37 +15.90 



-excess return 

""'number of observations 

" " "t - Stat is t i c 



16 



The paper by Asquith, Bruner and Mullins deserves additional 
comment. Schipper and Thompson (1983) have shown that acquisition 
programs generate excess returns. If this is true, one might expect 
the impact of successive mergers to diminish. Asquith et. al. compare 
the abnormal returns associated with the first, second, third, and 
fourth mergers. They find no evidence that abnormal returns are 
capitalized in the earlier mergers. They also found that the abnormal 
returns to the acquiring firm is dependent on the size of the acquired 
firm. 



Options 



Pricing 



The seminal work on option pricing is, of course, the Black- 
Scholes option pricing model. Black and Scholes (1973) noted that a 
call and the underlying stock could be combined to form a risk-free 
hedged portfolio if continuous rebalancing was possible. This fact, 
combined with some appropriate assumptions 

1. frictionless capital markets 

2. risk-free interest rate is constant 

3. stock pays no dividends 

A. stock prices follow an Ito process with constant drift 

5. no restrictions on short sales 
allowed them to derive a differential equation relating call and stock 
prices. Using stochastic calculus, they solved for the call price 
yielding the familiar Black-Scholes formula as a result 

C = SN(d^) - Xe-^T^J(^i^) 
where 



17 



d;, = [ln(S X) + (r + 0.5o-)T]/ajT 
d2 = dT_ - aVT 

The most limiting of the Black-Scholes restrictions is that the 
underlying stock pays no dividends. Modifying the model for dividends 
has two components. First, the stock price must be adjusted for the 
expected drop on the ex-dividend date. Second, the model must reflect 
that an American call has value due to its early exercise right. If a 
dividend is large enough, it may pay to exercise the option immediately 
before the stock goes ex-dividend. These problems can be dealt with 
simply by subtracting the present value of future dividends from the 
stock price as Black (1975) has suggested or assuming that dividends 
are paid continuously as Merton (1973) has done. Roll (1977), Geske 
(1979b), and Whaley (1981) have advanced more complex formulation that 
take both considerations into account. Whaley (1979) has empirically 
tested the different approaches to dividend adjustment and found the 
differences were slight. 

A number of variants and extensions of the Black-Scholes model 
have appeared. Merton (1973) has relaxed the assumption of stationary 
interest rates. Thorpe (1973) has examined the effect of short sales 
restrictions. Geske (1979a) has developed a compound option formula. 

The effects of different distributional assumptions regarding 
stock prices have also been investigated. Cox and Ross (1976) have 
developed a pure jump model that allows for discrete stock price 
movements. They have also developed a constant elasticity of variance 
model that allows for the variance to change with the stock price. 
Merton (1976) has developed a mixed diffusion- jump model that super- 
imposes a jump process on a continuous return generating process. 



18 



Despite these advances, the Black-Scholes model with the stock 
prices adjusted for dividends is still the most widely used by far. 
Many of the models discussed above are difficult if not impossible to 
apply. Even if they can be applied, no model has yielded consistent, 
significantly better results for options near the money. While there 
are limitations to the Black-Scholes model, there is no strong reason 
to use any of the more esoteric alternatives in this study. 

Option Market Efficiency 

A number of studies have been made of the efficiency of the 
Chicago Board of Options exchange. These studies are joint tests of 
market efficiency and the Black-Scholes model. Galai (1977) conducted 
one of the earliest and most comprehensive studies using the Black- 
Scholes model to identify mispriced options. He found that statisti- 
cally (but not economic) significant excess returns could be earned. 

Chiras and Manaster (1978) adopted a different approach in 
analyzing option market efficiency. They weighted the ISD of each 
option on the stock by the option price elasticity to arrive at an 
overall measure of the stock's future volatility (WISD) . They then 
compared the WISD as an estimate of future stock volatility to esti- 
mates based on past stock returns. Having demonstrated the superiority 
of WISDs, they then proceed to compute imiplied option prices. Under- 
priced and overpriced options were then identified by comparing implied 
and actual prices. Risk-free hedges were then formed which earned 
substantial abnormal returns. These results are in agreement with a 
similar study by Trippi (1977) which used a simpler weighting scheme to 
arrive at WISDs. Kalay and Subrahmanyara (1984) have also provided some 



19 

evidence of option market inefficiency on the ex-dividend day of the 
equity. 

Phillips and Smith (1980) have found fault with studies reporting 
inefficiencies in the option market. They argued that a close examina- 
tion of trading costs (most notably the bid-ask spread) would account 
for the abnormal returns reported in earlier studies. Bhattacharya' s 
(1980) study of CBOE (Chicago Board of Options Exchange) took these 
costs into consideration. In general, his results were consistent with 
market efficiency. 

Variance Bias in the Black-Scholes Model 

In order to apply the Black-Scholes model, five input variables 
must be obtained: the stock price, exercise price, time to maturity, 
risk-free rate of interest and the volatility of the underlying stock. 
Of these, four are directly observable. Only the variance of the 
underlying stock returns needs to be estimated. 

Classical methods of estimating the variance will bias the model. 
Although unbiased estimators of the variance exist, the Black-Scholes 
model is highly non-linear. Equal deviations from the true variance 
will not result in equal deviations from the true call price as 
Ingersoll (1977) and Merton (1975) have observed. Boyle and 
Ananthanrayanan (1977) have used numerical integration to examine the 
magnitude of the expected error in a single case. Butler and Schachter 
(1986) trace the behavior of this bias to the second derivative of the 
cumulative normal density function. 



CHAPTER 3 
THE BEHAVIOR OF OPTIONS AND OPTION MARKETS 
AROUND MERGER AND ACQUISITION ANNOUNCEMENTS 



This chapter discusses the behavior of options and option markets 
around merger and acquisition announcements. The impact of merger and 
acquisition announcements will be studied by examining the behavior of 
option prices and ISDs around the announcement date. These results 
will be compared to those obtained from the underlying equity using the 
traditional event study methodology. 

The organization of this chapter is as follows. First, the data 
is described and a potential problem discussed. Next, the behavior of 
options around merger and acquisition announcements is analyzed. The 
return of an option is affected by two company specific variables: the 
stock price and stock volatility. Both of these variables are likely 
to be affected by merger and acquisition activity and thus influence 
call returns. An attempt is made to decompose the total impact of 
merger and acquisition activity into two components. First, the effect 
of changing ISDs is investigated and then the total impact due to both 
changing stock prices and stock volatility is analyzed. Finally, the 
traditional event study methodology is applied to the underlying stock 
in order to compare the behavior of the two markets. 

Data 
The merger and acquisitions selected for this srudy will be 
obtained from Mergers and Acquisitions . The mergers and acquisitions 



20 



21 



selected will be confined to those involving at least $100 million in 
assets with either the acquiring or acquired firm having options listed 
on the CBOE between 1982 and 1985. The reason for this is to ensure 
the merger and acquisition is an event. Corporations listed on the 
CBOE tend to be well established firms with large equity bases. The 
value of all the outstanding stock in firms such as General Motors, 
General Electric and International Business Machines, for example, is 
measured in the billions of dollars. The announcement date will come 
from the Wall Street Journal Index. 

The Wall Street Journal will also be used to get the bid-asked 
spread on U.S. Treasury bills in order to calculate the risk-free 
rate. The risk-free rate for input into Black-Scholes formula will be 
the yield on the T-bill maturing closest to the expiration date of the 
option. The yield will be calculated according to the formula below 
from Cox and Rubinstein (1985, p. 255) 

r = (F/lG,GOO)'l/t 

where r = one plus the risk-free rate 

P = price of a $10,000 T-bill 

= 10,000 [l-0.01(bid + asked)/2 (n/360)] 

n = number of days to maturity 

t = time to maturity expressed in years 

The critical data for this thesis is the stock and option prices. 
Closing prices from the Wall Street Journal or similar sources can not 
be used because of the possibility of nonsynchronous trading between 
the two markets. Trading in the option market is significantly less 
active than in the equity market. It is quite likely that the last 



22 



option trade occurred prior to the last stock trade on any given day. 
The current stock price for use in the Black-Scholes formula is the 
price existing at the time the option is being valued. Using the 
Black-Scholes model to generate predicted returns as in this study 
requires the stock price at the time of the last option trade. This 
problem, as Bookstaber (1981) has pointed out, casts doubt on much of 
the empirical work on options that has been done to date. 

In order to avoid any problems with nonsynchronous trading, 
time-stamped data will be used. The Berkeley Options Tape will be the 
primary source of stock and option price information. Data from 
Francis Emory Fitch, Inc., although not machine readable, is also 
suitable. 

The Behavior of Options Around Merger 
and Acquisition Announcements 

The behavior of stocks around merger announcements has been 
extensively studied (see chapter 2). The results have been very 
consistent. Most of the gain due to merger activity is captured by the 
stockholders of the target firm. Gains to the bidding firm share- 
holders are small and possibly non-existent. 

These results suggest empirically testable hypotheses for the 
expected behavior of options around merger and acquisition announce- 
ments. Market anticipation of formal merger announcements has been 
observed in the equity markets. Under the assumption that the option 
market is efficient, option prices and ISDs should react prior to the 
formal merger and acquisition announcement and stabilize immediately 
afterward. A second hypothesis is that the abnormal returns to the 
target firm optionholders should exceed those of the bidding firm 



23 



optionholders. Theoretically, a call option may be duplicated by an 
appropriately selected stock-bond portfolio. Because of this, the 
wealth effects of merger and acquisition announcements can be expected 
to mirror that of the equityholders. This assumes, however, that other 
factors such as the stock volatility are not affected by the merger and 
acquisition activity. 

There are two main reasons why an analysis of the impact of merger 
and acquisition activity of optionholders wealth is of interest. 
First, the methodology used, the traditional event study, has never 
been applied to the option market. An event study in the option 
market presents new issues and sheds light, as will be discussed later, 
on event studies in the equity market. The second major reason why the 
wealth effects of merger and acquisition activity is important is that 
options afford superior leverage to the underlying equity by their very 
nature. Optionholders, per dollar invested, have more reason to be 
concerned with the effects of merger and acquisition activity than the 
equityholders . 

These ideas will be more fully developed later on. At this point, 

i 

1 the effect of merger and acquisition on option ISDs will be discussed. 

The behavior of ISDs around merger and acquisition announcements is not 

I only a major determinant of call option returns and thus optionholders 

wealth but also has important implications for event studies in the 

equity market . 

Event studies in the equity market implicitly assume that risk 

1 remains constant. Predicted returns are based on historical data from 

some base period. Increasing ISDs (i.e., stock volatility) would 

suggest that risk is increasing. More importantly, since ISDs can be 



24 



computed at a point in time and are correlated with stock betas, they 
can be used to adjust for increasing risk in the event period. This is 
discussed later on in this chapter. 

The Behavior of ISPs Around Merger 
and Acquisition Announcements 

The behavior of option ISDs around merger and acquisition 
announcements is important for a variety of reasons. First, it is 
inseparable from the price behavior of options. Stock volatility is 
one of the input variables for the Black-Scholes model. By examining 
the behavior of the ISDs it is possible to decompose changes in option 
prices into a component due to price changes in the underlying stock 
and a component due to changes in the underlying volatility. 

A second reason for examining the behavior of ISDs is that it 
provides an alternative measure of the information content associated 
with merger and acquisition announcements. The vast majority of event 
studies have attempted to measure the information content of some event 
by showing the expected return of the stock is affected. Patell and 
Wolf son (1979,1981) have pointed out that other moments of the stock 
price distribution may also be affected and thus serve as a measure of 
its significance. They used ISDs as an ex-ante measure of the informa- 
tion content associated with earnings announcements whose date is 
known . 

This study will determine if the second moment (stock volatility) 
of the stock return distribution is affected by merger and acquisition 
activity. The behavior of ISDs will be tracked around the announcement 
date for both the bidding and target firms in order to determine if 
there is a difference in the impact of the activity between the two 






25 



categories. It should be noted that merger and acquisition announce- 
ments are unexpected or at best partially anticipated. This fact 
distinguishes this study from the ones by Patell an Wolfson which dealt 
with earnings announcements on known dates. 

Another reason for examining the behavior of ISDs is that it may 
shed light on potential wealth shifts engendered by merger and acquisi- 
tion activity. Option theory suggests that common stock can be 
interpreted as an option. Agency theory suggests that there is an 
incentive for stockholders (see Jensen and Heckling 1976) to shift 
wealth from the bondholders by undertaking risky investm.ent projects. 
By undertaking investment projects which increase the variability of 
the firm's cash flows, the stockholders' can, in effect, gamble with 
the bondholder's money. This enriches the stockholders at the direct 
expense of the bondholders. Merger and acquisition activity can be 
regarded just like any other investment activity. Consequently, one 
might expect bidding firms to make acquisitions which tend to increase 
the variability of the firm's cash flows. 

Others, however, have argued that the opposite occurs. Levy and 
Sarnat (1970), Lewellen (1971) and Galai and Masulis (1976) have argued 
that combining the cash flows of two independent com.panies may reduce 
the probability of default and increase the market value of debt at 
the stockholders' expense. Even if this occurs, it is possible that 
managers act to neutralize (issue more debt) any such wealth shift. 

In any case, it is the variability of the firm's cash flow that is 
in question. A direct relationship, however, has been hypothesized in 
previous work (see Eger 1983). Consequently, the behavior of ISDs 



26 



around merger/acquisition announcements may be of value in analyzing 

whether these wealth shifts do, in fact, take place. 

■Methodology 

The methodology used for analyzing ISD behavior is as follows. 
First the sample was stratified into two groups. The first group was 
composed of 52 bidding firms involved in a merger or acquisition. The 
second group was composed of 21 target firms involved in a merger or 
divestment. 

Base ISDs were obtained by "inverting" the Black-Scholes model 
using data forty days prior to the announcement date. It is assumed 
that the markets have not yet begun to reflect the merger and acquisi- 
tion activity at this point. If the 40th day prior to the announcement 
is a holiday or weekend, the first trading day afterwards is used. 
Dividends are assumed to be paid continuously and are adjusted for as 
suggested by Merton^ (1973). The stock and option prices are the first 
prices from the Berkeley option tapes after the stock price has changed 
once. The opening trade is eliminated in order to ensure the market 
has stabilized. ISDs are calculated in a similar manner for each 
company for each day in the event period. The event period ranges from 
five days prior to the announcement date to two days afterward. It 
should be noted that the announcement day is taken to be the date it 
first appeared in the Wall Street Journal . In many instances, the news 
was released during trading hours of the previous day. 



Dividends are adjusted for by using Merton's (1973) formula 
C = Se"ytN(di) - Xe'^^Nid-y) 
where dl = [ln(S/X^ + (r - y - 0T5a2)t]/aVt 
d2 = dl - oVt 
y = continuous dividend vield. 



27 



The impact of the merger and acquisition activity on option ISDs 
was measured by taking the difference between the ISD for each company 
for each day during the event period and the base ISD for each company 

SISDjt " ISDjt - ISDbj 

where 6ISD^^ = Change in ISD for company j on day t. 

(t = -5 to +2) 

ISD-^ = ISD for company j on say t 

ISDb^ = Base ISD for company j 

A t-test was run on the change in ISDs for each day in order to 
determine statistical significance: 

t = 6ISDj/(s2/N)"l/2 

The results are given in Table 3.1. For each day in the event 
period the mean change in the ISD is given, the t-statistic and the 
probability (if significant at the lOZ level) of exceeding the absolute 
value of the t-statistic given there was no change in the distribution 
of ISDs between the base and event periods. 

The effect of changing ISDs on call prices was also investigated. 
For each company, for each day during the event period, the closing 
price of the option closest to the money with at least 30 days to 
maturity was obtained. The Black-Scholes Model was used to compute a 
call price on the same option using the base ISD but actual (market) 
stock prices. The percentage difference between the actual (market) 
call price and the theoretical base price obtained using base period 
ISDs in the Black-Scholes model was calculated for each company for 
each day in the event period 



28 



Table 3.1 
Average Change in ISDs Between the Base and Event Period 



Bidding Firms 



ay 


Average 


t- 


Prob > 




ISD Change 


Statistic 


It! 


-5 


-0.010678 


-1.41 




-4 


-0.009359 


-1.46 




— s 


-0.002800 


-0.39 




-2 


0.000765 


0.12 




-1 


-0.001085 


-0.17 







-0.005835 


-0.73 




+1 


-0.005354 


-0.58 




+2 


0.000932 


0.10 





Target Firms 



Day 


Average 


t- 


Prob > 




ISD Change 


Statistic 


/t/ 


-5 


0.025402 


1.35 




-4 


0.021308 


1.34 




-3 


0.033090 


1.92 


.0690 


-2 


0.034044 


2.01 


.0580 


-1 


0.048052 


3.03 


.0071 





0.056132 


2.39 


.0266 


+1 


0.024384 


1.53 




+2 


0.004447 


0.24 





29 



where %&C^^^ = % Difference between the actual and the base 

call price for company j on day t 

C-!^ = Actual market call price for company j on day t 

Cb^|- = Base price obtained from using base ISD in the 
Black-Scholes Model for company j on day t. 



Since the observed (market) stock price is used to obtain the base call 
price (Cb^-j-), the difference between the actual and base call prices 
must be entirely due to the changing stock volatility. 

A t-test was run on the percentage deviation from the actual call 
prices in order to determine statistical significance 



t = %6Cjt / (s2 / N)"l/2 



The results are given in Table 3. For each day in the event period, 
the mean percentage deviation is given, the t-statistic and the 
probability (if significant at the 10% level) of exceeding the absolute 
value of the t-statistic assuming there was no change between the base 
and actual market call prices. 
I-'-erpretation of Results 

As one might expect, the above two tables are very consistent. 
They may be regarded as opposite sides of the same coin. The change in 
ISDs for the bidding firm is small and statistically insignificant (at 
the 10% level) in all instances. Similarly, the percentage deviation 
of market prices from base prices is also small and statistically 
insignificant. The change in ISDs for the target firm are much larger 
than those of the bidding firm for corresponding days in the event 
period. Furthermore, the change is always positive and statistically 



30 



significant for days -3 through the announcement date (Day 0). The 
same observations hold for the percentage deviation in prices. 

These results are consistent with the hypotheses of option market 
efficiency. For the bidding firm there is no evidence that the merger 
and acquisition activity has any effect on the volatility of the 
underlying stock. The change in ISDs are very small and do not result 
in large, statistically significant changes in the call prices. There 
does not appear to be any changes in the ISDs or call prices before and 
after the merger and acquisition announcement. The target firms are 
definitely affected by the merger and acquisition acrivity. The 
average change in ISDs is over 5 percentage points in absolute terms on 
Day and is responsible for call price increases of ovex 12%. The 
market, however, starts to anticipate the merger and acquisition 
announcement as early as three days ahead of time. The change in ISD 
from the base level jumps from roughly 0.021 on day -4 to 0.033 on day 
-3.0 to -0.048 on day -1 to 0.056 on day 0. The percentage change in 
call prices follow a similar pattern. Immediately after the announce- 
ment is made public, however, ISDs and call prices quickly stabilize at 
close to their base levels. The deviation of the market from the base 
call price is only 0.009 for the target firms the day after the 
announcement . 

The results in Tables 3.1 and 3.2 also support the hypothesis that 
ISDs can be used to measure the information content of merger and 
acquisitions announcements. Studies in the equity market have shown 
that mergers do not greatly affect the expected return of the bidding 
firm stockholders. It would appear that the volatility of returns is 



31 



Table 3.2 
Percentage Deviation Between Market and Base Call Prices 



Day 



Bidding Firms 
% Deviation t-Statistic 



Prob > /t/ 



-5 
-4 
-3 
-2 
-1 

+1 
+2 



-0,009791 
-0.016577 
-0.058152 
0.002362 
-0.030124 
■0.021935 
-0.021193 
■0.013851 



0.48 
-0.78 
-1.11 

0.12 
-1.49 
-1.05 
-0.89 
-0.55 



Target Firms 



Day 


% Deviation 


t- Statistic 


Prob > Itl 


-5 


0.035339 


0.76 




-4 


0.037151 


0.85 




-3 


0.067024 


1.66 


0.1133 


-2 


0.085629 


1.74 


0.0966 


-1 


0.126755 


3.14 


0.0056 





0.058533 


1.65 


0.1146 


+1 


0.008906 


0.10 




+2 


-0.047437 


-0.83 





32 



also unaffected. Changes in the bidding firm ISDs are very small and 
statistically insignificant. 

Studies in the equity market have also shown that significant 
abnormal returns accrue to the target firm shareholders. The results 
here indicate these abnormal returns are accompanied by increased 
return volatility. Tt should be noted that the numbers in Table 3.1 
are absolute changes from the base ISD. The percentage deviations from 
the base ISD would be much larger. 

Why does merger and acquisition activity have such a major impact 
of the second moment (variance) of the return distribution of the 
target firms? As mentioned earlier, results in the equity market have 
shown that most of the gain from merger activity is captured by the 
target firm shareholders. The rationale for this is that the takeover 
market is competitive. If a company has some unique aspect that other 
companies can exploit, it will find or have the potential to find a 
number of bidders. Competition among the bidding firms will drive the 
net present value of the investment to zero (see Mandelker (197A) and 
Jensen and Ruback (1983)). Consequently, the gains from merger and 
acquisition activity will be reaped by the target firm shareholders. 

Because of this, merger and acquisition activity could be expected 
to affect the volatility of the target firm's equity much more then 
that of the bidding firm's. Merger and acquisition activity is a more 
or less neutral event for the bidding firm shareholders. Target firm 
shareholders, however, are likely to be greatly affected. The impor- 
tance of merger and acquisition activity to the target firm share- 
holders combined with uncertainty over the terms of the agreement, 
whether alternative bidders will appear, whether the agreement will be 



33 



consummated, etc., should result in higher ISDs for the target firm 
options. 

Table 3.2 confirms this hypothesis. The average change for the 
bidding firms is negligible. The average change in ISD for the 
target firms is larger than that of the bidding firms for the corre- 
sponding day in all cases. In some instances, the change in the 
target firms' ISDs exceeds those of the bidding firms by more than an 
order of magnitude. 

This result warrants further comment. Event studies in the 
equity market have demonstrated time and time again that target firm 
shareholders reap abnormal returns as a result of merger activity. 
These abnormal returns, however, are accompanied by increased vola- 
tility as Table 3.2 shows. Thus, these "abnormal returns" may not 
truly be abnormal but merely reflect the increased uncertainty and 
riskiness engendered by the merger and acquisition activity. Instan- 
taneous or short-term adjustments for risk are difficult in the equity 
market since beta requires historical time series to estimate. The ISD 
of an option, however, can be determined at a point in time. Thus, an 
event study in the option market may afford a better measure of excess 
return. This point is explored more deeply in the next section. 

A final comment on the behavior of ISDs. It is interesting that 
the ISDs for both the bidding and target firms' revert back to their 
base level after the merger and acquisition announcement (see Table 
3.2). It would appear that merger and acquisition activity does not 
result in permanent changes in the volatility of the underlying equity 
for either bidding or target firms. The evidence here does not support 
the hypotheses that wealth shifts between bondholders and stockholders 



34 



arises from merger and acquisition activity. This is counterintuitive. 

One would expect the post-announcement ISD to be function of the 

volatility of the underlying equity of both companies and their 

correlation. These results may be due to the sample selection process. 

Companies listed on the CBOE tend to be large com.panies. Thus, a 

merger and acquisition of $100 million may still be insignificant. In 

addition, large mergers or acquisitions are likely to take place 

between solid, established companies of relatively equal size. Thus, 

any conclusions concerning wealth shifts and merger and acquisition 

activity based on the data here must be interpreted with great care. 

The Behavior of Call Option Prices Around Merger and Acquisition 
Announcements 

Many of the reasons for examining the behavior of option prices 
around merger and acquisition announcements have already been discussed 
earlier. First, it provides a test of option market efficiency. 
Merger (and acquisition) negotiations involve many people such as 
investment bankers, lawyers, administrative personnel, etc. Word of 
impending mergers leaking to the financial market place has been amply 
demonstrated in the equity market (see Keown and Pinkerton, 1981). 
There is no reason why the same phenomenon should not occur in the 
option market. 

The price behavior of options around merger and acquisition 
announcements is important to anyone who intends to invest or speculate 
in options. Merger and acquisition activity is a major economic factor 
in our economy and is likely to remain so for some time. Anyone 
involved in options may be confronted with an unanticipated merger or 
acquisition announcement. Options by their very nature afford superior 
leverage to the underlying equity. Optionholders, per dollar invested. 



35 



are more affected by merger and acquisition activity than the equity- 
holders. In order to invest intelligently, potential optionholders (or 
sellers) need to have some idea of how merger and acquisition activity 
could potentially affect their wealth position. 

Another reason for analyzing the behavior of option prices around 
merger and acquisition announcements is that it may help to determine 
if this type of activity is an "event" from the standpoint of the 
bidding firm. As noted earlier, gains to the bidding share holders are 
small, possibly negative and statistically insignificant (see Chapter 
2). It also appears that merger and acquisition activity has no effect 
on the volatility (second moment of the return distribution) of the 
bidding firm's equity. It is possible, however, that the option 
prices of the bidding firms might still measurably react to the merger 
and acquisition activity. 

An option can be interpreted as a leveraged position in the 
equity. The leverage aspect of options may make them more sensitive to 
events than the underlying equity. A shock or event that provides an 
insignificant abnormal return in the equity market might be magnified 
into an identifiable, significant abnormal return in the option market. 
Thus, it might be easier from a statistical standpoint to determine if 
merger and acquisition activity is an event to the bidding firm 
security holders. 

A final reason for examining the price behavior of call options is 
that this is the first study to apply the traditional event study 
methodology to the option market. The event analyzed here is merger 
and acquisition announcements. The methodology employed, however, has 



36 



general applicability. It can be applied to any event such as dividend 
or earning announcements. 
Methodology and results 

An event study attempts to measure the impact of some event on 
securityholders by comparing the actual, observed market returns to 
those predicted by some model. Ideally, these predicted returns should 
be the returns that would have occurred if the event (merger and 
acquisition activity) had not taken place. This study uses the Black- 
Scholes model to generate predicted returns. 

In a Black-Scholes framework, call option prices change when 
there is a change in the risk-free rate, the time to expiration, the 
exercise price, the stock price or stock volatility. In equilibrium, 
the actual, observed market price equals the theoretical, Black-Scholes 
price. Here the announcement effect is measured by the impact of the 
changing stock price and volatility on the option price. The observed 
call option price is compared to a predicted price generated by the 
Black-Scholes model that keeps the stock price and volatility constant. 

An event study in the option market is fundamentally different 
from the one in the equity market. Event studies in the equity market 
assume the return generating process is linear and that the true beta 
remains constant over time. As long as predicted returns equal actual 
returns on average, residuals should average out to zero over a large 
enough cross-sectional sample in the absence of some common disturbing 
event. This also justifies using an estimate of beta. The true beta 
is unobservable and must be estim.ated. If an unbiased estimate of beta 
is used, deviations from the true expected return will also offset and 



37 



residuals should average out to zero in the absence of a common 
disturbance. 

The return generating process in the option market, however, 
makes an event study inherently different from one in the equity 
market. A cursory examination of the Black-Scholes model shows it is 
highly non- linear. Even if unbiased estimators are used to obtain 
inputs for the model, equal deviations from the true parameters will 
not result in equal deviations from the true call price. Thus, 
residuals will be biased simply due to the estimation of the input 
variables. This issue has important implications for users of the 
Black-Scholes model and is analyzed at length in Chapter 4. 

Another difference between an event study in the two markets is 
that the uncertainty of an option is an explicit function of time. The 
uncertainty of an option with a short time to maturity is greater than 
the same option with a longer life. The Black-Scholes model incor- 
porates this time dependency and will be used for this study. 

The methodology used to examine the behavior of call prices 
around merger and acquisition announcements is as follows. First, as 
before, the sample was divided into bidding and target firms. A 
number of options with different exercise prices and maturities exist 
for a company on a listed exchange. One option was selected to avoid 
statistical dependence in the returns. The exercise price selected was 
the closest price to the stock price forty days prior to the announce- 
ment. It is assumed that the impending announcement will not be 
reflected in option prices at this point. The maturity selected will 
be the first expiration date at least thirty days after the event 
period. 



38 



The reason for these particular choices of exercise price and 
maturity is to mitigate problems with using the Black-Schoies formula. 
The Black-Scholes formula has been found to be less accurate for deep 
in-the-money or out-of-the-money options. Thus, an option near-the- 
money is used. The reason for insisting the option have at least 30 
days to expiration is that the Black-Scholes model has been shown (see 
Manaster and Rendleman 1982) to be sensitive to its underlying assump- 
tions for options close to expiration. 

Once an option is selected according to this criteria, returns 
will be computed for each day in the event period. These returns will 
be matched with predicted returns computed from prices generated by the 
Black-Scholes formula. 

The predicted returns should be untainted by the merger and 
acquisition activity. Of the five input variables for the Black- 
Scholes model, only the stock volatility and stock price are likely to 
be affected. The obvious approach to estimating the stock volatility is 
to use historical stock returns from some base period. Another method 
is to "solve" the Black-Scholes formula for the implied standard 
deviation. ISDs reflect market expectations and should provide better 
estimates of future stock volatility than historical data. This has 
been confirmed by Latane and Rendelman (1976), Trippi (1977) and Chiras 
and Manaster (1978). Although a number of complex weighting schemes 
have been suggested, Beckers (1981) has demonstrated that using the ISD 
from the option nearest the money may work just as well. For this 
reason, the base ISDs computed in the previous section will be used to 
proxy the base volatility. The efficient market hypothesis suggests 
that the best estimate of tomorrow's stock price is today's stock 



39 



price. For this reason, the closing stock price 40 days prior to the 
announcement date (which is assumed to be unaffected by merger and 
acquisition activity) is used as the input stock price. 

Residuals will be computed for each company for each day in the 
event period 



"j.t = Rj,t - R'^j,t 



where U^,^ = residual for company j on day t 



J 
R 



j,|- - actual (observed) option return for company 
j on day t 



R*jjt ~ predicted option return for company j on day t 

Next daily average residuals will be com.puted to measure the impact of 
the merger and acquisition announcement for each day in the event 
period 

_ n 

U^- = 1/N .E,U. ^ 
^ j=i J>t 

where U*. = average daily residual for day t 
N = number of observations 

Finally, cumulative average residuals (CARs) will be calculated to 
measure the total abnormal return accruing to the optionholders. 



t 
CAR^ = S U^ 

t=-4 



The statistical significance will be measured by a t-test on 
the daily residuals 



40 



^'t ' -^^ 



7 n (u. ^ - e) 



1 



. , N - 1 



Residuals for the bidding and target firm will, of course, be treated 
separately. The results are given in Table 3.3. For each day in the 
event period, the daily average residual, t-statistic, probability (if 
significant) of exceeding the absolute value of the t-statistic and 
CAR are given. 
Interpretation of results 

It is interesting to note that with the possible exception of the 
bidding firms' behavior on day-2, the results in Table 3.3 are consis- 
tent with the results in Tables 3.1 and 3.2. Merger and acquisition 
activity has a much larger impact on the target firm option holders 
than the bidding firm optionholders. The cumulative average residual 
is about 7.5% through the announcement day for the bidding firm options 
versus about 39% for the target firm. Abnormal returns for the target 
firm options are statistically significant two days and the day before 
the announcement. 

It would seem, however, that merger and acquisition activity is an 
event for the bidding firm option holders. The excess return of 6.4% 
two days before the announcement is highly significant. This is 
consistent with the ISD behavior of the bidding firms' options on day- 
2. Although not statistically significant, the ISD does change sign 
and become positive (see Table 3.1). The issue of whether merger or 
acquisition, should be regarded as an event (having measurable impact) 



41 



Table 3.3 



Abnormal Returns in the Option Market 
Around Merger and Acquisition Announcements 



Bidding Firms 





Daily 








Day 


Average 


t- 


Prob > 


Car 




Residual 


Statistic 


Itl 




Dav-4 


0.004993 


0.21 




0.004993 


Day-3 


0.007201 


0.37 




0.012194 


Day-2 


0.064110 


2.87 


0.0059 


0.076304 


Day-1 


0.015931 


0.40 




0.092235 


Day 


-0.017628 


-0.62 




0.074607 


Day+1 


0.019649 


0.74 




0.094256 


Day+2 


0.022679 

Tar 
Daily 


0.92 
get Firms 




0.116935 


Day 


Average 


t- 


Prob > 


Car 




Residual 


Statistic 


Iti 




Day-4 


-0.015935 


-0.52 


. 


-0.015935 


Day-3 


0.033639 


0.69 




0.017704 


Day-2 


0.128971 


1.75 


0.0956 


0.146675 


Day-1 


0.210493 


2.68 


0.0152 


0.357168 


Day 


0.031266 


0.45 




0.388374 


Day+1 


0.017990 


0.51 




0.406364 


Day+2 


0.025705 


-0.90 




0.380659 



42 



on the bidding firm optionholders will be returned to in the next 
section. 

The above results, as might be expected, are consistent with the 
hypothesis of market efficiency. For both the bidding and target 
firms, the formal announcement is anticipated. After the merger and 
acquisition is made public, there are no excess returns. 

The "abnormal returns" in Table 4 are based upon the traditional 
event study methodology that has been used in the equity market. That 
is, the parameter(s) (beta in the equity market) for the model generat- 
ing the predicted returns are estimated using data from some base 
period free from the disturbing effects of the event (merger and 
acquisition) activity. The difference between the actual, observed 
market returns and the predicted returns is defined to be the excess or 
abnormal return. 

This excess return assumes that the risk (beta) does not change. 
In actuality, merger and acquisition activity may not benefit a 
stockholder even if abnormal returns are observed. These abnormal 
returns may be accompanied by increased risk engendered by the merger 
and acquisitionn activity. If risk were compensated for on a contin- 
uous basis, it is possible that the abnormal returns reported would 
disappear. This has not been done in the equity market since estimat- 
ing beta requires time series data over a relatively lengthy period of 
time. The issue is explored more fully in the next section. 

For an event study in the option market, it is not necessary to 
estimate beta. The relevant counterpart is the stock volatility for 
which the ISD can be used as a proxy. The ISD, however, unlike beta, 
can be computed at a point in time. This allows for a more complete 



43 



current estimate of predicted returns for event studies in the option 
market than in the equity market. 

In order to demonstrate this, the event study above was rerun for 
the target firms on Day -2 and Day -1 (which yielded abnormal returns). 
The only difference is that ISDs from the previous day (rather than 40 
days prior to the announcement date) were used in the Black-Scholes 
model to generate predicted returns. That is, prices for day-3 were 
based on ISDs from day-4, prices for day-2 were based on prices from 
day-3 and prices for day-1 were based on ISDs from day-2. All other 
aspects of the study are identical. The results are shown in Table 
3.4. 

The implication of these results is that abnormal returns reported 

in event studies to date may be overstated. Using the previous day's 

ISDs to reflect a more current measure of the stock volatility reduced 

the excess return on day-2 by almost three percentage points. Although 

there is no direct relation between a stock's volatility and beta, it 

would seem logical that merger and acquisition activity could have 

short run effects. If beta could be observed on a continuous basis so 

that equity returns could be properly adjusted for risk, abnormal 

returns might be substantially reduced or even eliminated. This is 

discussed in more detail in the next section. 

The behavior of option markets around merger and acquisition 
announcements 

This section extends the event study in the option market 

to the underlying equity. The reason for doing this is to compare the 

behavior of the two markets around the announcement of merger and 

acquisitions. There are two major reasons for doing this. 



Table 3.4 

Target Firm Option Abnormal Returns 
Based On Previous Day ISDs 



Daily 

Day Average t- Prob > 

Residual Statistic t 

Day-2 0.099975 1.45 0.1617 

Day-1 0.208600 2.49 0.0288 



45 



The first is that it places the option market results in perspec- 
tive. While the absolute level of abnormal returns is of interest in 
itself, it is important to compare the level of excess returns in the 
option and equity markets. An investor concerned with merger and 
acquisition activity would need to know the relative effects before he 
could properly allocate his resources between the two markets. 

The second reason for extending the event study to the underlying 
stocks is that the two markets may behave differently. There are two 
independent arguments for the hypothesis that merger and acquisition 
activity will be first manifested in the option (rather than equity) 
market. 

Options can be interpreted as leveraged positions in the underly- 
ing equity. The beta of an option is always greater than that of the 
underlying asset (stock). Thus, it is possible that the option market 
may be more sensitive to events than the equity market. In other 
words, although both markets may have received the same bit of informa- 
tion, the signal may be "magnified" and first apparent in the option 
market . 

It is also possible that the option market contains information 
that is not incorporated in the equity market prior to major corporate 
announcements. As mentioned previously, a call or put option can be 
duplicated by an appropriate stock-bond portfolio. Because of this, 
options have been viewed as "derivative" assets whose prices are 
completely determined by the underlying equity. The possibility that 
the option market may influence the equity market has received little 
attention. Information may first be processed in the option market and 
then filter to the equity market. 



46 



This issue has been investigated by Manaster and Rendleman 
U982). They advanced the intriguing hypothesis that the option 
market may play a key role in determining equilibrium stock prices. 
They argue that some investors may prefer to invest in the option 
rather than the equity market because of reduced transaction costs, 
fewer short selling restrictions and most importantly, superior 
leverage. These traders could push option prices out of equilibrium 
relative to the underlying stocks. Arbitragers would then intervene 
to restore equilibrium between the two markets. 

Manaster and Rendleman attempted to test their theory. They 
"inverted" the Black-Scholes model to solve for the implied stock 
price. The implied stock price was then used to predict future stock 
prices. They found some evidence that the option market contains 
information that is not incorporated in the equity market. Unfortu- 
nately, their results are very weak and fatally flawed by their 
reliance on non-synchronous data. The data used in this dissertation 
avoids this problem. 

In retrospect, Manaster and Rendlemans' lack of results is 
not surprising. Both the option and equity markets react to public 
information. Generally, one would expect both markets to adjust 
simultaneously to new public information. On any given day for any 
particular corporation there may not be and probably is not information 
that is not fully reflected in both markets. 

However, this may not be true prior to m.ajor announcements by 
corporations such as mergers or acquisitions. In this case, the option 
market could be expected to be particularly influential in determining 
stock prices. Keown and Pinkerton (1981) have argued that information 



47 



concerning impending mergers is susceptible to insider exploitation due 
to the large number of people typically involved in the negotiating 
process. They have attributed increased trading volume before the 
merger announcement to insider activity. An insider attempting to 
profit from knowledge of an impending merger would have an incentive to 
use options because of their leverage aspects. To quote Fischer Black 
(1975, p. 61), "Since an investor can usually get more action from a 
given investment in options than he can be investing in the common 
stock, he may choose to deal with options when he feels he has an 
especially important piece of information." Option prices can be 
expected ro contain more information than the equity market if non- 
public information is being exploited. If information regarding future 
mergers first reaches the financial markets through insider trading in 
options, merger activity may very well be reflected in the option 
market before the stock market. 
Methodology and results 

The standard event study methodology was applied to the equity 
market. Daily returns for each day in the event period were obtained 
from the Center for Security Price Research (CRSP) tapes. These 
observed market returns were then compared to mean returns . Mean 
returns were computed using returns for the sixty trading days prior to 
the base date forty days prior to the announcement date. 

Residual computation and analysis is as before. Residuals are 
calculated for each company for each day in the event period 



^'j't ~ ^i,t ' '^i 



where U- ^ = residual for company j on day t 



48 



where U4 ^ = residual for company i on day t 

R-j ^ = actual equity return for company j on day t 
R-j = mean return for company j 

Next daily average residuals are com.puted to measure the impact of the 
merger or acquisition announcement for each day in the event period 



_ n 

U^ = 1/n E U.^. 
3 = i 



Cumulative average residuals are also calculated to measure the total 
excess return accruing to the equityholder . 

Statistical significance of the residuals is measured as before by 
a t-test on the daily residuals. 



U, • Vn 
t = 



n (U - e )2 

v' E -^ r^- 

n - i 

j = l 



The results are given in Table 3.5. For each day in the event period, 
the daily average residual, t-statistic and probability of exceeding 
the absolute value of the t-statistic, if significant, is given. 

Table 3.5 is consistent with other merger studies done in the 
equity market. Merger and acquisition activity has very little impact 
on the bidding firms. The largest daily average residual, although 
statistically significant at the 10% level is only 0.0044. For the 
target firms, a statistically significant daily average return of 
almost 0.04 was observed on day-1. 

The abnormal returns for the target firm equityholders is a 
little low compared to returns obtained in other merger studies. This 



49 



Table 3.5 



Abnormal Returns In The Equity Market 
Around Merger and Acquisition Announcements 



Bidding Firms 



Day 



Day-4 
Day-3 
Day-2 
Day-1 
Day 
Day+1 
Day+2 



Daily 

Average 

Residual 

-0.001457 

0.001275 

0.004015 

-0.002133 

-0.007647 

-0.003259 

0.000562 



t- 


Prob > 




atistic 


/t/ 


Car 


-0.65 




-0.001457 


0.50 




-0,000182 


1.71 


0.0926 


0.003833 


-0.52 




0.001700 


-2.15 


0.0362 


-0.005947 


1.26 




-0.009206 


0.22 




-0.008694 



Target 





Daily 






Average 


t- 


Day 


Residual 


Statistic 


Dsy-4 


-0.006512 


-1.68 


Day-3 


-0.002720 


-0.50 


Day-2 


0.007597 


1.07 


Day-1 


0.039875 


+2.44 


Day 


0.000620 


0.08 


Day+1 


-0.002300 


-0.36 


Day+2 


0.000402 


0.08 



Prob I 
/t/ 



0.0240 



Car 

-0.006512 
-0.008932 
-0.001335 
0.038540 
0.039160 
0.036860 
0.037262 



50 



concerning impending mergers is susceptible to insider exploitation due 
to the large number of people typically involved in the negotiating 
process. They have attributed increased trading volume before the 
merger announcement to insider activity. An insider attempting to 
profit from knowledge of an impending merger would have an incentive to 
use options because of their leverage aspects. To quote Fischer Black 
(1975, p. 61), "Since an investor can usually get more action from a 
given investment in options than he can be investing in the common 
stock, he may choose to deal with options when he feels he has an 
especially important piece of information." Option prices can be 
expected to contain more information than the equity market if non- 
public information is being exploited. If information regarding future 
mergers first reaches the financial markets through insider trading in 
options, merger activity may very well be reflected in the option 
market before the stock market. 
Methodology and results 

The standard event study methodology was applied to the equity 
market. Daily returns for each day in the event period were obtained 
from the Center for Security Price Research (CRSP) tapes. These 
observed market returns were then compared to mean returns. Mean 
returns were computed using returns for the sixty trading days prior to 
the base date forty days prior to the announcement date. 

Residual computation and analysis is as before. Residuals are 
calculated for each company for each day in the event period 



"j't = Rj,t - Rj 



51 



is probably due to the sample. Companies listed on the CBOE tend to be 
large, established companies. The takeover market may not be as 
efficient for firms of this size. Relatively few companies have the 
resources to undertake an acquisition of this scope. This fact is 
reflected in the sample. Of the 21 target firms in the sample, seven 
are mergers. For the divestitures, abnormal returns may also be 
comparatively small due to the size of the firms involved. Although 
large in absolute terms, a $100 million divestiture for a company such 
as General Electric is likely to have very little impact. 

Although the pattern of abnormal returns is similar for both 
markets, the residuals in the option market tend to be much larger. 
For the bidding firms, cumulative average residuals were 0.074607 
through the announcement day for the options vice -0.005947 for the 
equity. For the target firms, cumulative average residuals were 
0.388374 and 0.039160 for the options and equity, respectively. 

The only puzzling feature in the above tables is the statis- 
tically significant excess return for the bidding firms options 
observed two days prior to the announcement date. The results in the 
equity market, however, are consistent. The average residual for day- 
2, although small in absolute terms, is large compared to those of 
other days and is statistically significant. It should be noted that 
the data source used in the option and equity markets are independent. 
The Berkeley option tapes served as the basis for the option event 
study and the CRSP tapes for the equity. 

The results for day-2 are also not due to low priced options. A 
small price change on an option priced at less than a dollar could 
result in large returns that m.ight not actually be realizable. This 



52 



possibility was checked for by redoing the analysis for the bidding 
firm options on day-2 and the target firm options on day-1. This 
time, however, rerurns based on prices less than one dollar are 
eliminated. The results are given in Table 3.6. The daily average 
residual for the bidding firm does decline from about 6.4% to roughly 
A%. It is, however, still statistically significant. Eliminating the 
low priced options from the target firms actually increases the daily 
average residual. 

The results from this study support the hypotheses that merger and 
acquisition activity is first manifested in the option market. For the 
bidding firms in the equity market, the daily average residual is 
uniformly small. In the option market, however, there is a large jump 
between the daily average residual of 0,007201 on day-3 and 0.06AI10 on 
day-2. For the target firms, the evidence is more pronounced. In the 
equity market, the merger and acquisition activity is not evident until 
day-1. In the option market, the merger activity is definitely 
reflected by the excess returns on day-2 and arguably on day-3. The 
target firm ISDs, however, have started to react three days prior to 
the announcement. 

As noted earlier, abnormal returns were obtained for the target 
firms in the equity market (see Table 3.5). These abnormal returns, 
however, were based on historical data. Thus, an underlying assumption 
is that the risk (beta) does not change. In reality, merger and 
acquisition activity may be accompanied by increased risk that is not 
reflected in the base beta. If beta could be observed on a continuous 
basis so that equity returns could be properly adjusted for risk, 
abnormal returns might be substantially reduced or even eliminated. 



53 



Table 3.6 



Selected Abnormal Returns with Call Prices 
Under $1.00 Eliminated 

















Bidding Fii 
Day-2 


■ms 




Target Firms 
Day-1 


Daily 

A/erage 

Residual 


t- 
Statistic 


Prob > 
Itl 


Daily 

Average 

Residual 


t- 
Statistic 


Prob > 
Itl 


0.039593 


1.98 


0.0536 


0.248540 


2.76 


0.0147 



54 



A number of attempts were made to adjust for risk in the equity 
market by exploiting the high correlation between a stocks volatility 
and beta. Unlike beta which requires time-series data, the ISD can be 
calculated at a point in time. Although there is no theoretical 
relationship between the ISD and a stock's beta, empirical relation- 
ships can be established. These relationships can then be used to 
adjust for the increasing risk due to the impending merger or acquisi- 
tion announcement. 

The methodology used to adjust the laevel of risk in the equity 
market during the event period involves regressing stock betas against 
their volatility. Daily returns for the target firms were regressed 
against the market (CRSP value weighted) index for the six months 
prior to the base (AO days prior to the announcement) date. This 
yielded the intercept for the market model and an unadjusted beta to 
conduct an event study in the equity market. Stock volatilities based 
on the daily returns were also calculated. The target firm betas were 
regressed against the volatilities to obtain the following relation 

B = 0.866764 + 19.48914--'' a R^ = 0.098 

A similar relationship was obtained using annual data. Annual 
returns for the target firm were regressed against the market (CRSP 
value weighted) index for the thirty years 1952 to 1981. Stock 
volatilities based on this annual data were also computed. The annual 
betas were then regressed against the stock volatilities to obtain 

B = -0.061901 + 3.669232-- a R^ = 0.714 



55 



These relationships were used to adjust the beta for each day in 
the event period. The ISD was plugged into the equations above to 
obtain an adjusted beta. These adjusted betas were then plugged into 
the market model (based on the daily returns) to generate predicted 
returns. The standard event study methodology was then used to obtain 
the results shown in Table 3.7. The first colmnn is the residuals and 
associated t-statistics obtained from using an unadjusted beta, that 
is, the beta based on the six months of daily data. The second column 
shows the results obtained when the ISD is used to adjust the beta 
using the relationship between beta and a based on the annual data. 
The third coliamn shows the results when the ISD for each day in the 
event period is used to adjust the beta using the relationship based on 
the daily data. 

These results indicate the adjustments for risk were not success- 
ful. On day-1, the abnormal return are almost identical regardless of 
whether the unadjusted beta, adjusted beta based on annual return data 
or daily return data is used. Either the adjustment procedure is 
flawed or the level of risk did not change during the event period. An 
analysis of the data reveals a technical reason why the adjustment 
procedure did not work. 

In a CAPM framework, stocks must have an expected return greater 
than the risk-free rate. Ex-post, however, negative returns do occur. 
Many of the market returns on the day prior to the announcement date 
(day-1) were negative in this sample. The average market return is 
-0.000808. The practical effect of this is that adjusting beta upwards 
can result in larger abnormal returns (residuals) because of the data. 
If the market return is negative, increasing beta will only result in a 



56 



Table 3.7 



Abnormal Returns for the Target Firms 
for Various Beta Adiustments 



Day Unadjusted Annually Daily 

Beta Adjusted Adjusted 

Beta Beta 

-4 -0.002448 -0.003664 -0.003431 

(t = -0.59) (t = -0.89) (t = -0.81) 

-3 -0.002102 -0.001601 -0.001539 

(t = -0.51) (t = -0.41) (t = -0.39) 

-2 0.005530 0.005232 0.005516 

(t = 0.84) (t = 0.79) (t = 0.82) 

-1 0.032000 0.033300 0.033147 

(t = 2.34) (t = 2.35) (t = 2.34) 

Q 0.001017 0.001048 0.001032 

(t = 0.13) (t = 0.14) (t = 0.14) 

+1 -0.002665 -0.002597 -0.002729 

(t = -0.66) (t = -0.64) (t = 0.26) 

+2 0.001301 0.000929 0.001190 

(t = 0.28) (t = 0.19) (t = 0.26) 



57 



lower predicted return. This illustrated by the abnormal returns that 
result from the following adjustment to beta for day-1 

Bg = B[(ISD(-1) - ISDb)/lSDb) + 1.0] • k 

where B^ = adjusted beta 

B = base beta obtained from six months daily data 

ISD(-l) = ISD on day-1 

ISDb = base ISD 

k = an arbitrary scaler 

The results for k = 1, 1,3, 1.5, and 2.0 are shown in Table 3.8. Here 
we see that increasing beta has very little impact on the residuals. A 
larger beta results in a larger predicted return (smaller residual) 
impact for those companies for which the market return was positive. 
This is offset, however, by those companies for which the market return 
is negative. 

The magnitude by which beta would have to be increased in order to 
eliminated the abnormal returns can still be calculated. Adding 0.015 
to the market returns on day-1 to make them positive and B-'.OIS to the 
company returns does not change the residuals but makes the adjustment 
process conform to theoretical expectations. The above regressions 
were rerun with the indicated adjustment. The results are given in 
Table 3.9. These results show that adjusting the base beta by the 
percentage change in the ISDs times a scaler of 1.40 reduces the 
abnormal returns to statistical insignificance. This suggests that the 
basic methodology used to adjust beta above is sound but needs to be 
applied to a larger sample where the average market return is positive. 



58 



Table 3.8 



Abnormal Returns Obtained by Adjusting Beta by the 
Percentage Change in ISDs Times a Scaler (K) 



K = 1.0 K = 1.3 K = 1.5 K - 2.0 



Day -1 0.033378 0.033793 0.034070 0.034751 
(t - 2.34) (t = 2.35) (t = 2.35) (t = 2.33) 



59 



Table 3.9 



Abnormal Returns Obtained by x'Vdjusting beta by the 

Percentage Change in the ISD and a Scaler (K) 

After Adding 0.015 to the Market Returns and 

B"'.015 to the Company Returns 



K = 1.0 K = 1.10 K = 1.20 K = 1.30 K = 1.40 

Day-1 0.029318 0.027321 0.025323 0.023325 0.021327 

(t = 2.19) (t = 2.05) (t = 1.91) (t = 1.77) (t = 1.62) 

(0.0422) (0.0552) (0.0720) (0.0939) (12.17) 



CHAPTER 4 
VARIANCE BIAS AND NON- SYNCHRONOUS PRICES 
IN THE BLACK- SCHOLES MODEL 



One of the underlying assumptions of an event study in the equity 
market is that the return generating process is linear. As long as 
predicted returns equal observed (actual market) returns on average, 
residuals (abnormal returns) should also average out to zero. It is 
the common disturbance (event) that generates abnormal returns. 

This linearity of the return generating process also justifies 
using an estimate of beta. If an unbiasec estimator of beta is used, 
errors will tend to offset. The estimates of beta may be high or low 
but will average out in a large sample. Furthermore, the error in 
estimated returns and thus residuals will also average out to zero. 

The Black-Scholes model, however, is highly nonlinear. Thus, 
using an estimate for the input variables may result in a systematic 
bias. Even if an unbiased estimator is used for the input variables 
(most notably the stock volatility), errors from the true call price 
will not offset even in a large sample. The reason for this is that 
equal deviations from the true input parameters will not result in 
equal deviations from the true call price. 

This has implications far beyond that of conducting an event study 
in the options market. Applying the Black-Scholes model has an 
inherent bias due to the fact that the formula is non- linear and input 
variables must be estimated. The magnitude and direction of these 



60 



61 



biases is of interest to any user of the Black-Scholes model. For this 
reason, the issue of bias in the Black-Scholes model arising from these 
sources is considered in a broader context rather than as a technical 
issue concerning event study methodology. 

This chapter has two sections. The first section deals with the 
bias that results in the Black-Scholes model from using a sample 
estimate of the variance with all other input parameters assumed to be 
known. The following section extends the analysis to uncertainty in 
the underlying stock price due to non-synchronous trading (or price 
quotes) between the option and equity markets. 

Variance Bias in the Black-Scholes Model 
The Black-Scholes model is by far the most widely used option 
pricing formula. In order to apply it, five input variables must be 
obtained: the stock price, exercise price, time to maturity, risk-free 
rate of interest and the volatility of the underlying stock. Of these 
variables, four are directly observable. Only the variance of the 
underlying stock returns needs to be estimated. Hull and White (1987) 
have analyzed the impact of cp- itself being stochastic on the call 
option value. In this paper, however, we assume that a^ is constant 
but its estimate, S'^, is a random variable. 

Classical methods of estimating the variance will bias the Black- 
Scholes model as Ingersoll (1975) and Merton (1976) have pointed out. 
To see this, define Z^ = Ln(l + R^) where R|. is the rate of return on 
the underlying stock in period t and assijme that Z^ is an independent, 
normally distributed random variable. The unbiased estimate, a^, of 
the variance of the stock returns is given by 



62 





n 

t=i 


- Z)2 


§2 = 


= 





(4.1) 

N - 1 



where N is the number of observations 



_ n 
Z = S Z^/N 
t=l 



While it is well known that S'^ is an unbiased estimate of a^, it is not 
true that E(C) = C where C is the value derived from the Black-Scholes 
formula with the true but unknown a^ and C is a random variable 
calculated by employing the Black-Scholes formula with the random 
variable S^. 

Let us elaborate this point. The Black-Scholes model is given by 

C = SgNCd^) - Ee'^tfj(^i2) ^ ^_2 

where d^ = [InCSg/E) + (r + 0.5a2)t]/aVt 
d2 = dj_ - aVt 

and the sample estimate of C is given by C 

C = SQN(d^) - Ee"^%(d2) ^-3 

where dj^ = [InCSg/E) + (r 4- 0.5s2)t]/sV t 
d2 = dj_ - sVt 

(recall that S is a random variable) 

It is obvious that E(C) ^ C for the following reasons. First, even 
if E(s2) = a2, E(S) ^ a and a is one of the inputs into the Black- 
Scholes formula. Second, even if E(S) = a (which it does not), E(C) ^ 



63 



C since a appears in the denominator of the formula and E(l/S) ^ l/o. 
Even if E(S) = a and E(l/S) = l/o, the model would still be biased due 
to its non-linearity. Equal deviations from the the true a^ would not 
result in equal deviations from the true option price. 

Analyzing the gap between E(C) and C is difficult. One has to 
evaluate the following difference 

LnCSg/E) + (r+0.5s2)/sVt LnCSg/E) + (r-0. 5S)t/s2Vt 

■O.SZ^ -rt -0.5Z2 



E(C) - C = Sg J 1/V2TT e dz - Ee J 1/V2tt 



e dz 



LnCSo/E) + (r+0.5a2)t/aVt Ln(Sn/E) + (r-0.5o2)t/aVt 



-0.5Z2 -rt -0.5Z2 

-Sq J 1/V2TT e dz + Ee J 1/V2tt e dz 



A closed-form solution to the first two integrals is extremely complex 
since S, a random variable, appears in the upper bound. Boyle and 
Anathanarayanan (1977) used numerical integration to approximate the 
above integrals and investigated the case of an option expiring in 90 
days. 

In this paper, we provide an alternative approach by using 
simulation. Sample estimates, S'-, of the stock volatility, o" , 
are generated and used to compute option prices using the Black- 
Scholes formula. These prices are then compared to the theoretical 
value determined by using the true o" in the Black-Scholes formula in 
order to measure the bias induced. This is repeated for options with 



64 



various maturities. The dispersion of sample call prices from the 
theoretical value is also investigated. 

Methodology and Results 
The effect of using a sample estimate of the variance in the 
Black-Scholes model was analyzed using simulation analysis. For this 
purpose, an option with the following characteristics was chosen. 
These parameters were representative of IBM options in the early 
1980 's. Note that the true stock volatility is assumed known. 

Stock price = $68,125 

Risk-Free Rate = 0.1325 
of Interest 

True Standard = 0.4472 
Deviation of 
Stock Returns 

Time to Maturity = Various 

Exercise Price = Various 

The simulation is based on the well known relationship 
'iTi'.veen the sample and true variance 

9 

S is distributed as a Chi-square with N-1 degrees of freedom which for 
this analysis is assumed to be fifty-nine. This implies that the 
sample variance was estimated using sixty observations. One thousand 
Chi-square deviates were obtained using the International Mathematical 
and Statistical Library (IMSL) computer program. The sample variance 
was then computed for each Chi-square observation for input into the 
Black-Scholes formula. For each exercise price, one thousand call 



65 



prices using the sample variances obtained from simulation were 
calculated and the average computed. Options with five, sixty, and two 
hundred seventy days to maturity were examined. 

The results are given in Table 1. The theoretical price assxaming 
the true variance is known is given for each exercise price. The 
exercise price changes in five percent increments from the given stock 
price of $68,125. The average simulation price is the mean of the 
thousand generated call prices. The percentage bias is calculated by 

% bias = '^'^h^o^^^tical price-average simulation price) * 100 

theoretical price 

Note that a positive bias is associated with average simulation prices 
less than the theoretical Black-Scholes prices. 

These results show a definite bias exists. While mean simulated 
prices do deviate from theoretical Elack-Scholes prices, however, the 
differences are small. In most cases, the average simulation price is 
within a few cents of the theoretical price. The largest difference is 
approximately eight cents. The percentage bias is also sm.all. For the 
options with 60 and 270 days to expiration, it is always under one 
percent. While biases over one percent do occur for the option with 
five days to maturity, they are at prices so low as to be economically 
meaningless . 

A few observations on the nature of the bias between the average 
simulation value and the theoretical value should be made. First, A 
downward bias exists in most cases. The average value obtained from 
simulation is less than the theoretical value for all nine exercise 
prices for the options with sixty and two hundred seventy days to 
expiration. For the five day option, the theoretical price exceeds the 



66 



Table 4.1 
Theoretical Call Price and Average Simulation Call Price 





T = 5 Days 


to Maturity 




Exercise 


Theoretical 


Average Simu- 


Percent 


Price 


Price 


lation Price 


Bias 


5A.500 


13.7178 


13.7157 


0.01523 


57.906 


10.3184 


10.3175 


0.00872 


61.313 


6.9418 


6.9427 


-0.01253 


64.719 


3.7987 


3.7966 


0.05503 


58.125 


1.4799 


1.4701 


0.66084 


71.531 


0.3669 


0.3636 


0.89129 


74.938 


0.0549 


0.0568 


-3.49790 


78.344 


0.0050 


0.0060 


-20.16123 


81.750 


0.0003 


0.0005 


-50.71423 




T = 60 Days to Maturity 




Exercise 


Theoretical 


Average Simu- 


Percent 


Price 


Price 


lation Price 


Bias 


54.500 


15.1807 


15.1806 


0.00040 


57.906 


12.3118 


12.3038 


0.06482 


61.313 


9.7241 


9.7056 


0.19035 


64.719 


7.4747 


7.4468 


0.37366 


68.125 


5.5917 


5.5583 


0.59802 


71.531 


4.0735 


4.0402 


0.81773 


74.938 


2.8926 


2.8639 


0.99112 


78.344 


2.0055 


1.9846 


1.04311 


81.750 


1.3596 


1.3474 


0.90314 




T = 270 Days to Maturity 




Exercise 


Theoretical 


Average Simu- 


Percent 


Price 


Price 


lation Price 


Bias 


54.500 


20.9884 


20.9614 


0.12868 


57.906 


18.7896 


18.7513 


0.20132 


61.313 


16.7597 


16.7107 


0.29198 


64.719 


14.9000 


14.8419 


0.39013 


68.125 


13.2070 


13.1416 


0.49526 


71.531 


11.6748 


11.6044 


0.60293 


74.938 


10.2950 


10.2220 


0.70870 


78.344 


9.0591 


8.9858 


0.80924 


81.750 


7.9562 


7.8849 


0.89591 



67 



average simulation price for five exercise prices. In the other four 
cases, the bias is extremely small amounting to less than one cent. 

The bias is largest in absolute terms for the options with longer 
maturities. However, there is no systematic relationship when the bias 
is expressed in percentage terms. When the bias is expressed in 
percentage terms, the bias for the sixty day option is smaller than 
that of the two hundred seventy day option at low exercise prices but 
larger at high relative exercise prices. 

For an option of a given maturity, the bias is more pronounced at 
high exercise prices. This holds true regardless of whether the bias 
is expressed in absolute or percentage terms. This makes intuitive 
sense. At low exercise prices most of an option's value is due to its 
intrinsic worth. At high exercise prices more of the option's value 
can be attributed to the volatility of the underlying stock. Conse- 
quently, the estimate of the variance becomes more important. 

The results in Table 4,1 are encouraging to users of the Black- 
Scholes model. The bias in a large sample is small. This does not 
guarantee, however, that using a sample estimate of the variance will 
not severely degrade the applicability of the Black-Scholes model. 
Sample call prices might each differ from the theoretical price by a 
great amount. In a large sample these individual errors might offset 
so that the average error was small. The dispersion of the sample call 
prices from the theoretical value is also crucial. 

For this reason, average simulated prices were generated on the 
same IBM option as before only using A, 6, 8, 10, 15 and 30 runs 
instead of a thousand. The results are given in Table 4.2. As before, 
options with 5, 60 and 270 days to expiration were examined. For each 



68 



Table 4.2 
Average Simulated Values and Bias for Various Sample Sizes 



Maturity = 5 Days 
Exercise Price = $54.50 
Theoretical Price = $13.7178 



Number 


of 


Average 


Simulat 


ions 


Simulation Price 


4 




13.7178 


6 




13.7178 


8 




13.7178 


10 




13.7178 


15 




13.7178 


30 




13.7178 



Percent 
Bias 

0.0000 
0.0000 
0.0000 
0.0000 
0.0000 
0.0000 



Maturity = 5 Days 
Exercise Price = $68,125 
Theoretical Price = $1,4799 



Number 


of 


Average 


Simulat 


ions 


Simulation Price 


4 




1.5181 


6 




1.4626 


8 




1.4516 


10 




1.4526 


15 




1.4608 


30 




1.4669 



Percent 
Bias 

-2.5812 
1.1690 
1.9123 
1.8448 
1.2906 
0.8785 



Maturity = 5 Days 
Exercise Price = $81,750 
Theoretical Price = $.0003 



Number 


of 


Average 


Percent 


Simulat: 


Lons 


Simulation Price 


Bias 


4 




0.0005 


-66.6667 


6 




0.0004 


-33.3333 


8 




0.0003 


0.0000 


10 




0.0003 


0.0000 


15 




0.0003 


0.0000 


30 




0.0004 


-33.3333 



69 



Table A. 2 (continued) 



Maturity = 60 Days 
Exercise Price = $54.50 
Theoretical Price = $15.1807 



Number 


of 


Average 


Simulat 


ions 


Simulation Price 


4 




15.2367 


6 




15.1680 


8 




15.1528 


10 




15.1526 


15 




15.1618 


30 




15.1742 



Percent 
Bias 

-0.3689 
0.0837 
0.1838 
0.1851 
0.1245 
0.0428 



Maturity = 60 Days 

Exercise Price = $68,125 

Theoretical Price = $5.5917 



Numbe 


r of 


Average 


Siinul 


ations 


Simulation Price 


4 




5. 7214 


6 




5.5328 


8 




5.4955 


10 




5.4980 


15 




5.5266 


30 




5.5473 



Percent 
Bias 

-2.3195 
1.0533 
1.7204 
1.6578 
1.1642 
0.7940 



Maturity = 60 Days 
Exercise Price = $81.75 
Theoretical Price = $1.3596 



Number of 
Simulations 



Average 

Simulation 

Price 



Percent 
Bias 



4 

6 

8 

10 

15 

30 



1.4633 
1.3239 
1.2949 
1.2962 
1.3159 
1.3355 



-7.6272 
2.6258 
4.7588 
4.6632 
3.2142 
1.7725 



70 



Table 4.2 (continued) 



Maturity = 270 Days 
Exercise Price = $54.50 
Theoretical Price = $20.9884 



Number 


of 


Average 


Simulat 


ions 


Simulation Price 


4 




21.1675 


6 




20.9231 


8 




20.8727 


10 




20.8753 


15 




20.9101 


30 




20.9429 



Percent 
Bias 

3.9112 
0.3112 
0.5513 
0.5389 
0.3731 
0.2168 



Maturity = 270 Days 
Exercise Price = $68,125 
Theoretical Price = $13.2070 



Number 


of 


Average 


Simulat 


ions 


Simulation Price 


4 




13.4635 


6 




13.0913 


8 




13.0177 


10 




13.0244 


15 




13.0789 


30 




13.1198 



Percent 
Bias 

-1.9421 
0.8760 
1.4333 
1.3826 
0.9699 
0.6603 



Maturity = 270 Days 
Exercise Price = $81.75 
Theoretical Price = $7.9562 



Number 


of 


Average 


Simulat 


ions 


Simulation Price 


4 




8.2374 


6 




7.8293 


8 




7.7485 


10 




7.7559 


15 




7.8157 


30 




7.8606 



Percent 
Bias 

-3.5344 
1.5950 
2.6105 
2.5175 
1.7671 
1.2016 



71 



of these maturities, exercise prices of $54.50, $68,125 and $81.75 were 
selected. The percent bias is calculated as before 5. The theoretical 
Black-Scholes price is also given for each option. 

These results show that the dispersion of option prices from their 
theoretical values due to using the sample variance is not great. The 
largest absolute difference is about $0.25. In general, the percentage 
bias is usually less than 2%. The major exception is for the out-of- 
the-money option with five days to maturity. This is due to the 
insignificant theoretical call prices (less than $0.01) associated with 
this option. 

The same observations concerning the behavior of the bias for the 
large sample (1000 runs) experiments apply to the small sample experi- 
ments. The bias is generally positive (theoretical price exceeds 
average simulated price). When the bias is negative, it is almost 
always associated with the smallest number of simulations (4). Again, 
the percentage bias is usually smallest at low exercise prices and 
becomes larger as the exercise price is increased. 

Non-synchronous Prices and the Black-Scholes Model 

Many investment decisions involving options are based on closing 
stock and option prices or other non-sychronous sources of data. Since 
the option market is much thinner than the stock market, these prices 
are often based on trades from different times of the day. The stock 
price prevailing at the time of the last option trade may be signifi- 
cantly different from the closing price at the end of the day. Conse- 
quently, using this stock price in the Black-Scholes model may cause 
options to appear mispriced as Trippi (1977), Chiras and Manaster 
(1978), Galai (1977) and Bookstaber (1981) have pointed out. 



72 



Here the raispricing of options that can occur due to the non- 
simultaneity or stock and option quotations and using a sample estimate 
for the variance of the underlying stock returns in the Black-Scholes 
model is examined. An option is constructed for analysis and its 
theoretical value is calculated assuming the input variables including 
the relevant stock price and true volatility a^, are known. This value 
is compared to call prices generated with the same parameters (includ- 
ing the true assumed volatility) only varying the input stock price 
from the assumed true stock price in order to measure the effects of 
nonsimultaneous stock and option quotations. 

The additional bias resulting from using a sample estimate of the 
variance is measured by simulation analysis. For each stock price, 
sample estimates, 3--, of the stock volatility, o^ , are generated and 
used to compute option prices using the Black-Scholes formula. These 
prices are then compared to the theoretical price determined by using 
the true variance, o^, and true (synchronous) stock price in order 
measure the bias due to the combination of the two factors. The 
methodology and results are describe below. 

Methodology and Results 
Simulation analysis was used to m.easure the effects in the 
Black-Scholes model of using a sample estimate of the variance in 
conjunction with nonsimultaneous stock and option quotations. For this 
purpose, an option with the following characteristics was chosen. These 
parameters are representative of a typical option traded on the Chicago 
Board Option Exchange in the mid 1980 's. Note that the true stock 
volatility and stock price are known. 



73 



Stock Price = $50,000 

Risk-Free Rate = 0.1000 
of Interest 

True Standard = 0.3500 
Deviation of 
Stock Returns 

Time to Maturity = Various 

Exercise Price = $50,000 

The effect of nonsiir.ultaneous stock and option prices alone on the 
Black-Scholes model was measured by varying input stock price in 1/8 
increments from the true stock price of $50,000. For each stock price 
between $49,000 and $51,000 the Black-Scholes value was computed using 
the parameters listed above including the true assumed variance of 
0.3500. 

The combined effects of nonsimultaneous price quotations and using 
a sample estimate of the variance was analyzed by simulation. The 
simulation is based on the relationship between the sample and true 
variance didcusssed earlier 

2 

S N . 1 ^ A. 4 

S^ is distributed as a Chi-square with N-1 degrees of freedom which for 
this analysis is assumed to be twenty-nine. This implies that the 
sample variance was estimated using thirty observations. One thousand 
Chi-square deviates were obtained using the International Mathematical 
and Statistical Library (IMSL) computer program. The sample variance 
was then computed for each Chi-square observation for input into the 
Black-Scholes formula. For each exercise price, one thousand call 
prices using the sample variances obtained from simulation were 



74 



calculated and the average computed. Options with five, sixty, and two 
hundred seventy days to maturity were examined. 

The results are given in Table 4.3. For each stock price, the 
Black-Scholes value is given based on the true variance of 0.3500. 
This gives a measure of the mispricing that can occur to nonsimul- 
taneous price quotations. For each of these stock prices, the average 
simulation price is also given. The average simulated price is the 
mean of the thousand generated call prices obtained with that exercise 
price and estimates of the variance. 

The percentage bias of these values from the theoretical value is 
also given. The percentage bias is calculated by 

% bias = ^^^^"^^^^^^^ price-average simulation price) * 100 

theoretical price 

Note that a positive bias is associated with average simulation prices 
less than the theoretical Black-Scholes prices. 

These results indicate that making investment decisions involving 
options on the basis of nonsynchronous price data must be made with 
great care. Even when the stock price is off by only an eighth the 
observed call price will deviate from its theoretical price by over one 
percent. For short maturities, using a stock price that deviates from 
the true stock price by one dollar can results in call prices that are 
over 50% off from the true value. The error due to using a sample 
estimate of the true variance is small in comparison to that caused by 
using nonconteraparenous stock prices. For stock prices above the true 
value, the two errors are offsetting. For stock prices below the true 
stock value, the two errors reinforce one another. 



75 



Table 4.3 



Mispricing in the Black-Scholes Model Due to Nonsimultaneous 
Stock and Option Quotations and Using a Sample Estimate 
for the Variance of the Underlying Stock Returns 



T = 5 Days to Maturity 
Theoretical Price = 0.843231 



Stock 


Black-Scholes 


Average 


Percent Bias 


Percent 


Price 


Price With 


Simulation 


Due to Non- 


Bias Due to 




True Variance 


Price 


Simultaneous 
Quotations 


Both Effects 


51.000 


1.455809 


1.452143 


-72.6465 


-72.2117 


50.875 


1.369338 


1.365278 


-62.3918 


-61.9103 


50.750 


1.285629 


1.281119 


-52.4646 


-51.9298 


50.625 


1.2046Q5 


1.199766 


-42.8558 


-42.2820 


50.500 


1.126509 


1.121297 


-33.5944 


-32.9762 


50.375 


1.051239 


1.045770 


-24.6679 


-24.0193 


50.250 


0.978912 


0.973244 


-16.0906 


-15.4184 


50.125 


0.909561 


0.903781 


-7.8662 


-7.1807 


50.000 


0.843231 


0.837410 


0.0000 


-0.6903 


49.875 


0.779953 


0.774134 


7.5042 


8.1943 


49.750 


0.719757 


0.713988 


14.6430 


15.3271 


49.625 


0.662537 


0.656968 


21.4288 


22.0892 


49.500 


0.608398 


0.603052 


27.8492 


28.4832 


49.375 


0.557251 


0.552217 


33.9148 


34.5117 


49.250 


0.509110 


0.504414 


39.6240 


40.1809 


49.125 


0.463882 


0.459599 


44.9875 


45.4954 


49.000 


0.417516 


0.500117 


50.0117 


50.4644 



76 



Table 4.3 (continued) 



T = 60 Days to Maturity 
Theoretical Price = 3.135864 



Stock 


Black-Scholes 


Average 


Percent Bias 


Percent 


Price 


Price With 


Simulation 


Due to Non- 


Bias Due to 




True Variance 


Price 


Simultaneous 
Quotations 


Both Effects 


51.000 


3.726805 


3.708286 


-18.8446 


-18.2540 


50.875 


3.649992 


3.631251 


-16.3951 


-15.7975 


50.750 


3.574020 


3.555066 


-13.9724 


-13.3680 


50.625 


3.498869 


3.479729 


-11.5759 


-10.9655 


50.500 


3.424468 


3.405258 


-9.2033 


-8.5907 


50.375 


3.351058 


3.331654 


-6.8623 


-6.2436 


50.250 


3.278503 


3.258919 


-4.5486 


-3.9241 


50.125 


3.187067 


3.187067 


-2.2607 


-1.6328 


50.000 


3.135864 


3.116100 


0.0000 


0.6303 


49.875 


3.065887 


3.046021 


2.2315 


2.8650 


49.750 


2.996780 


2.976843 


4.4353 


5.0711 


49.625 


2.928482 


2.908547 


6.6132 


7.2489 


49.500 


2.861113 


2.841156 


8.7617 


9.3980 


49.375 


2.794615 


2.774673 


10.8821 


11.5181 


49.250 


2.728973 


2.709085 


12.9754 


13.6096 


49.125 


2.664169 


2.644415 


15.0419 


15.6719 


49.000 


2.600462 


2.580659 


17.0735 


17.7050 



T = 270 Days to Maturity 
Theoretical Value = 7.302824 



Stock 


Black-Scholes 


Average 


Percent Bias 


Percent 


Price 


Price With 


Simulation 


Due to Non- 


Bias Due to 




True Variance 


Price 


Simultaneous 
Quotations 


Both Effects 


51.000 


7.948135 


7.909295 


-8.8364 


-8.3046 


50.875 


7.866165 


7.827100 


-7.7140 


-7.1791 


50.750 


7.784515 


7.745256 


-6.5959 


-6.0584 


50.625 


7.703323 


7.663871 


-5.4842 


-4.9439 


50.500 


7.622467 


7.582793 


-4.3677 


-3.8337 


50.375 


7.541931 


7.502134 


-3.2742 


-2.7292 


50.250 


7.461836 


7.421933 


-2.1774 


-1.6310 


50.125 


7.382185 


7.342033 


-1.0867 


-0.5369 


50.000 


7.302824 


7.262574 


0.0000 


0.5512 


49.875 


7.223953 


7.183511 


1.0800 


1.6338 


49.750 


7.145445 


7.104862 


2.1550 


2.7108 


49.625 


7.067305 


7.026569 


3.2250 


3.7829 


49.500 


5.989562 


6.948733 


4.2896 


4.8487 


49.375 


6.912291 


6.871284 


5.3477 


5.9092 


49.250 


6.835295 


6.794231 


6.4020 


6.9643 


49.125 


6.758818 


6.717631 


7.4493 


8.0132 


49.000 


6.682646 


6.641387 


8.4923 


9.0573 



77 



Conclusion 



In empirical tests of the black-Scholes model, one normally 
employs ex-post estimates of o"^ since a^ itself is unknown. While the 
sample variance is an unbiased estimate of a", the derived option value 
(which is a random variable) is a biased estimate of the true Black- 
Scholes value. 

The effects of this bias were analyzed by simulation. The true 
variance was assumed to be known and sample estimates generated by 
using a Chi-square distribution. One thousand sample variances and 
their associated call prices were obtained in each case. The average 
call price was calculated and compared to the theoretical Black-Scholes 
value. This process was performed on options with various maturities 
and exercise prices. 

The results show that using a sample estimate for the variance in 
the Black-Scholes model results in a downward bias. The average 
simulation price was less than the theoretical price for all options 
with 60 and 270 days to maturity. For the 5 day option, the average 
simulation price was less than the theoretical price for 5 of the 8 
exercise prices. When an upward bias was observed, it was not economi- 
cally significant. The downward bias was also evident in the small 
sample experiments. Sample call prices were generated in the same 
manner previously described only using fewer trials. Average simula- 
tion prices were computed using 4, 6, 8, 10, 15 and 30 runs. The 
average call prices generated by simulation were usually less than the 
theoretical Black-Scholes price for six or more runs. The differences 
between the average call prices generated by simulation and the 



78 



theoretical values were small. The percentage biases were also small 
except for deep-out-of-the-money options close to expiration. 

The effect of non- synchronous prices was also investigated. If 
the input stock prices deviate from the true stock price by only 1/8, 
the mispricing ranged from roughly 1% for the 270 day option to 
approximately 7% for the 5 day option. The additional error due to 
using an estimate of the variance was relatively small. 



CHAPTER 5 
SUMMARY AND CONCLUSIONS 



This dissertation investigated the behavior of options around 
merger and acquisition announcements. A variation of the traditional 
event study methodology was applied to the option market in order to 
determine the abnormal returns accruing to the bidding firm and target 
firm optionholders. The event study was then extended to the under- 
lying equity and the results between the two markets compared. 

In both the equity and option market, the effect of merger and 
acquisition activity was most pronounced for the target firms. The 
cumulative average residuals for the bidding firms in the equity market 
through the announcement date were close to zero. For the target 
firms, they were close to 4%. The corresponding CARs in the option 
market were 7.5% and 38.8%, respectively. 

The abnormal returns for the target firms in the option market are 
surprisingly large. Abnormal returns accruing to the optionholder are 
over 10 times as large as those accruing to the equityholders . The is 
due is not only the leverage effect in options but the fact that the 
stock volatility is increasing as well. 

Merger and acquisition activity can be expected to have a larger 
impact on the volatility (second moment of the return generating 
function) of the target firms than of the equity firms. Event studies 
in the equity market have shown that most of the gains from merger 
activity are captured by the target firm shareholders. The rationale 



79 



80 



for this is that the takeover market is competitive. If a company has 
some unique aspect to exploit, it will find or have the potential to 
find a number of bidders. Competition among the bidding firms will 
drive the net present value of the investment to zero. 

Because of this, merger and acquisition activity should be 
expected to affect the volatility of the target firms' much more than 
the bidding firms'. Merger and acquisition activity is a more or less 
neutral event for the bidding firm shareholders. Target firm 
shareholders are much more likely to be greatly affected. The 
importance of merger and acquisition activity combined with uncertainty 
over the terms of the agreement, whether alternative bidders will 
appear, whether the agreement will be consummated, etc., should result 
in higher ISDs for the target firms. 

This hypothesis was confirmed. The change in the ISDs between the 
event period and base date for the bidding firms was not significant. 
The changes were small and statistically insignificant. The target 
firms, however, had large statistically significant changes in the 
ISDs. 

The effect of changing stock volatility on option prices was also 
examined. Option prices in the event period were compared to those 
using the Black-Scholes model using the current stock price but the 
base ISD. The results showed that changing stock volatility was an 
important factor in the abnormal returns reaped by the target firm 
optionholders. 

The results of this study also suggest that merger and acquisition 
activity is first reflected in the option market. The target firm 
ISDs started to increase and were statistically significant 3 days 



81 



before the announcement. Target firm option returns had started to 
increase and were statistically significant 2 days before the announce- 
ment. Target firm stock returns, however, did not significantly 
increase until the day prior to the announcement. Bidding firm option 
returns were economically and statistically significant two days before 
the announcement. Bidding firm stock returns were very small through 
the announcement date although they were statistically significant 2 
days before the announcement. 

These results have practical implications for investors. If 
someone anticipates a company is about to announce a merger or acquisi- 
tion, they would reap much greater returns by purchasing options rather 
than the stock. Furthermore, they would be substantially better off by 
purchasing the target firm option rather than that of the bidding firm. 

This dissertation analyzed two issues involving the event study 
methodology. The first was the proper adjustment for risk in the 
equity market. Predicted returns have usually been based on data from 
some base period. The traditional event study methodology, thus, 
implicitly assumes that risk remains constant. It is far more likely 
that risk is actually changing due to the event (merger and acquisition 
activity). Abnormal returns would thus be overstated. Empirical 
relationships between the ISD (stock volatility) and beta were 
developed. These relationships were then used to adjust beta during 
the event period. Although conceptually sound, the results were 
disappointing due to a technical factor. The market return for many of 
the companies was negative. This resulted in smaller predicted returns 
(larger residuals) when larger betas were plugged into the market 
model. 



82 



The second major issue involves event studies in the option 
market. The Black-Scholes model is non-linear. Unbiased estimators 
for the input variables will still bias the results since equal 
deviations from the true input parameter value will not result in equal 
deviations from the true call price. Simulation analysis was used to 
measure the magnitude of this effect. The results indicate that 
although caution must be used in interpreting the results of an event 
study that uses the Black-Scholes model to generate predicted returns, 
the error is usually small. 



ij REFERENCES 

; Asquith, R., "Merger Bids, Uncertainty, and Stockholder Returns," 

1 Journal of Financial Economics , April 1983, 51-83. 

j Asquith, R. , R. Bruner, and D. Mullins, "The Gains to Bidding Firms 

j from Merger," Journal of Financial Economics , April 1983, 121-139. 

j Beckers, S,, "Predictors of Future Stock Price Variability," Journal 

i of Banking and Finance . September 1981, 363-382. 

j Bhattacharya, Mihir, "Empirical Properties of The Black-Scholes Model 

■| Under Ideal Conditions," Journal of Financial and Quantitative 

Analysis , December 198G, 1081-1106. 

Black, F., "Fact and Fantasy in the Use of Options," Financial Analysts 
Journal , July-August 1975, 61-72. 

I 

j and M. Scholes, "The Pricing of Options and Corporate 

I Liabilities." The Journal of Political Economy , May/ June 1973, 

''' 637-659. 

I Bookstaber, R. , "Observed Option Mispricing and the Nonsimultaneity of 

'Ij Stock and Option Quotations," Journal of Business , January 1981, 

'' 141-155. 

Boyle, P.P. and A.L. Ananthanarayanan, "The Impact of Variance Estima- 
:, tion in Option Valuation Models," Journal of Financial Economics , 

December 1977, 375-387. 

Brown, S. and J. Warner, "Event Studies with Daily Returns," Journal of 
Financial Economics , June 1985, 3-31. 

Butler, J.S. and B. Schachter, "Unbiased Estimation of the Black- 
Scholes Formula," Journal of Financial Economics , March 1986, 341- 
357. 

Chiras, D. and S. Manaster, "The Informational Content of Option Prices 
and a Test of Market Efficiency," Journal of Financial Economics , 
June- September 1978, 213-234. 

Cox, J. and S. Ross, "The Valuation of Option for Alternative 
Stochastic Processes," Journal of Financial Economics , January- 
March 1976, 145-166. 

Cos, J. and M. Rubinstein, Options Markets . Englewood Cliffs, N. J. : 
Prentice-Hall, 1985. 

83 



84 



Dodd, Peter, "Merger Proposals, Management Discretion and Stockholder 
Wealth," Journal of Financial Economics , June 1980, 105-137. 

Eckbo, B. Espen, "Horizontal Mergers, Collusion, and Stockholder 
Wealth," Journal of Financial Economics , April 1983, 241-273. 

Eger, C, "An Empirical Test of the Redistribution Effect in Pure 
Exchange Mergers", Journal of Financial and Quantitative Analysis , 
December 1983, 547-572. 

Finnerty, J., "Insiders and Market Efficiency," Journal of Finance , 
September 1976, 1141-1148. 

Galai, D. , "Tests of Market Efficiency of the Chicago Board of Options 
Exchange," Journal of Business , April 1977, 167-197. 

, "A Survey of Empirical Tests of Option Pricing Models," in 

Option Pricing , ed. Menachem Brenner, pp. 45-80. Lexington, 
Mass.: D.C. Heath, 1983. 

. and R. Masulis, "The Option Pricing Model and the Risk Factor 

of Common Stock", Journal of Financial Economics , January-March 
1976, 53-81. 

Geske, R. , "The Valuation of Compound Options," Journal of Financial 
Economics . March 1979a, 63-81. 

Geske, R. , "A Note on an Analytical Valuation Formula for Unprotected 
American Call Options with Known Dividends," Journal of Financial 
Economics , December 1979b, 375-380. 

Hull, John and A. WTiite, "The Pricing of Options with Stochastic 
Volatilities", Journal of Finance , 1987, 42, No. 2, 281-299. 

Ingersoll, J., "A Contingent-Claims Valuation of Convertible Securi- 
ties," Journal of Financial Economics , May 1977, 289-322. 

Jensen, M. and W. Heckling, "Theory of the Firm: Managerial Behavior, 
Agency Costs and Ownership Structure," Journal of Financial 
Economics , October 1976, 305-360. 

Jensen, M. and R. Ruback, "The Market for Corporate Control: The 
Scientific Evidence." Journal of Financial Economics , April 1983, 
5-50. 

Kalay, A. and M. Subrahmanyam, "The Ex-Dividend Day Behavior of Option 
Prices," Journal of Business , January 1984, 113-128. 

Keown, A. and J. Pinkerton, "Merger Announcements and Insider Trading 
Activity: An Empirical Investigation," Journal of Finance , 
September 1981, 855-869. 



85 



Latane, H. and R. J. Rendleman, Jr., "Standard Deviations of Stock 
Price Ratios Implied in Option Prices," The Journal of Finance , 
May 1976, 369-382. 

Levy, H. and M. Sarnat, "Diversification, Portfolio Analysis and the 
Uneasy Case of Conglomerate Mergers," Journal of Finance , Sept. 
1970, 795-807. 

Lewellen, W.G., "A Pure Financial Rationale for the Conglomerate 
Mergers," Journal of Finance , May 1971, 521-5A5. 

Malatesta, Paul H. , "The Wealth Effect of Merger Activity and the 
Objective Function of Merging Firms," Journal of Financial 
Economics , April 1983, 135- 181. 

Manaster, S. and R. Rendleman. "Option Prices as Predictors of Equili- 
brium Stock Prices," Journal of Finance . September 82, 1043-1057. 

Mandelker, G. , "Risk and Return: The Case of Merging Firms," Journal 
of Financial Economics , December 1974, 305-335. 

Merton, R. , "Theory of Rational Option Pricing," Bell Journal of 
Economics and Management Science , Spring 1973, 141-183. 

. "Option Pricing When Underlying Stock Returns are Discon- 
tinuous," Journal of Financial Economics , January-March 1976, 
125-144. 

, "The Impact on Option Pricing of Specification Error in the 

Underlying Stock Returns, Journal of Finance , 31, No. 2, 333-350. 

Patell, J.M. and M.A. Wolfson, "Anticipated Information Releases 
Reflected in Call Option Prices," Journal of Accounting and 
Economics , August 1979, 117 140. 

, "The Ex ante and Ex post Prices Effects of Quarterly Earnings 

Announcements Reflected in Option and Stock Prices," Journal of 
Accounting Research , Autumn 1981, 434-458. 

Phillips, Susan M. and Clifford W. Smith, Jr., "Trading Costs for 
Listed Options: The Implications for Market Efficiency," Journal 
of Financial Economics , June 1980, 179-201. 

Reid, R.S., Mergers, Managers, and the Economy , New York: McGraw Hill, 
1968. 

Roll, R. , "An Analytic Valuation Formula for Unprotected American Call 
Options on Stocks with Known Dividends," Journal of Financial 
Economics , November 1977, 251-258. 

Schipper, K. and R. Thompson, "Evidence on the Capitalized Value of 
Merger Activity for Acquiring Firms," Journal of Financial 
Economics , April 1983, 85-119. 



86 



Thorpe, Edward 0., "Extensions of the Black-Scholes Option Model," 
Proceeding of the 39th Session of the International Statistical 
Institute , August 1973, 1029-1036. 

Trippi, R., "A Test of Option Market Efficiency Using a Random-Walk 
Model," Journal of Economics and Business , Winter 1977, 93-98. 

Weston, J. and K. Chung, "Some Aspects of Merger Theory," Journal of 
the Midwest Financial Association 12, Spring 1983, 1-33. 

Whaley, Robert E. , "On the Valuation of American Call Options on Stocks 
with Known Dividends," Journal of Financial Economics , June 1981, 
207-212. 



87 



BIOGRAPHICAL SKETCH 

James A. Yoder was born on June 18, 1953, at Fort Monmouth, New 
Jersey. He received his Bachelor of Science degree in mathematics in 
197A from the State University of New York at Albany. He then went on 
to obtain an M.A. in economics in 1975 from the same university. 

Mr. Yoder entered the Navy in the Nuclear Power Program. He 
served on board the U.S.S. Dwight D. Eisenhower and made one 
Mediterranean deployment. After leaving the Navy, he completed his MBA 
at the State University of New York at Albany before entering the Ph.D. 
program at the University of Florida. 



I certify that I have read this study and that in my opinion it 
conforms to acceptable standards of scholarly presentation and is fully 
adequate, in scope and quality, as a dissertation for the degree of 
Doctor of Philosophy. 



1 < 

u 



c/-"^..'^'^" 



1 



Haim Levy, Chairman-'' 
Walter J. Mather ly Professor of 
Finance 



T ce-rtify that I have read this study and that in my opinion it 
conforms to acceptable standards of scholarly presentation and is fully 
adequate, in scope and quality, as a dissertation for the degree of 
Doctor of Philosophy. / 




/t,... 



Roy Cium 

Professor of Finance, Insurance, 
and Real Estate 



I certify that I have read this study and that in my opinion it 
conforms to acceptable standards of scholarly presentation and is fully 
adequate, in scope and quality, as a dissertation for the degree of 
Doctor of Philosophy. 




Sanfor/'V. Berg ^^ 
Profei^'sor of Economics 



This dissertation was submitted to the Graduate Faculty of the Depart- 
ment of Finance, Insurance, and Real Estate in the College of Business 
Administration and to the Graduate School and was accepted as partial 
fulfillment of the requirements for the degree of Doctor of Philosophy- 



Dean, Graduate School 



December 1988